PKu E toeblitz-1.0/PKu Etoeblitz-1.0/decomp/PK CI**$toeblitz-1.0/decomp/decompToeplitz.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % John P Cunningham, 2009 % Modified by William Zhang, 2013 % % decompToeplitz() % % Invert a symmetric, real, positive definite Toeplitz matrix % using either inv() or the Trench algorithm, which % uses Zohar 1969, finding the log determinant along the way. % This is slightly different than % Algorithm 4.7.3 of Golub and Van Loan % "Matrix Computations," the 1996 version, because the Zohar % version produces the log determinant essentially for free, which % is often useful in cases where one actually needs the full matrix inverse. % % If Zohar or the MATLAB implementation of the Trench algorithm is called, % the inversion is done in O(n^2) time, which is considerably better % than the O(n^3) that inv() offers. This follows the Trench % algorithm implementation of Zohar 1969. See that paper for % all explanation, as the C code is just an implementation of % the algorithm of p599 of that paper. See Algorithm 4.7.3 of Golub and Van Loan % for an explanation of the MATLAB version of this algorithm, which is slower % due to for loops (slower than inv(), up to about n=300). % Thus, if the C can be used, always use it. % % This function also computes the log determinant, which is % calculated essentially for free as part of the calculation of % the inverse when Zohar is used. This is often useful in applications when one really % needs to represent the inverse of a Toeplitz matrix. If Zohar is not used, the computation % is reasonably costly. % % Usage: [ld, Ti] = decompToeplitz(T, logdetMode, [runMode]); % % Inputs: % T the positive definite symmetric Toeplitz matrix to be inverted % logdetMode if this is set to 1, we skip the building of the inverse for efficiency, % and only compute the log determinant. If it is 0, we still compute both % the log determinant and the inverse % runMode OPTIONAL: to force this inversion to happen using a particular method. This % will NOT roll over to a method (by design), so it will fail if something is amuck. % % Outputs: % ld the log determinant of T, log(det(T)) % Ti the inverted matrix, also symmetric (and persymmetric), NOT TOEPLITZ % % NOTE: We bother with this method because we % need this particular matrix inversion to be % as fast as possible. Thus, no error checking % is done here as that would add needless computation. % Instead, the onus is on the caller to make sure % the matrix is toeplitz and symmetric and real. % % NOTE: Algorithm 4.7.3 in the Golub book has a number of typos % Do not use Alg 4.7.3, use the equations on p197-199. % % Run time tests (testToeplitz) suggest that Zohar is the fastest method across % the board. (2-3 orders of magnitude at sizes from 200-1000). Classical Trench is 13/4n^2 flops, % whereas inv() is probably 2/3n^3. However, the overhead for the MEX calls, etc., % may make the crossover point very MATLAB dependent. % % NOTE: The interested numerical linear algebra geek can dig into all this stuff by % running that testToeplitz to find the crossover % point him/herself. Some tinkering is required. This will run three inversion methods: inv(), decompToeplitzFastZohar (MEX Zohar), % decompToeplitz (Matlab Zohar). This version of decompToeplitz has been pared down to % be more user friendly, so it only includes a call to C/MEX Zohar, inv(), and vectorized MATLAB Zohar. % % MEX NOTE: This function will call a MEX routine. % A try block is included to default back to the best non-MEX solver (either inv() % or a vectorized Trench/Zohar version in native MATLAB), so hopefully the user will % not experience failures. We have also included the compiled .mex in a number of % different architectures (hopefully all). However, the user should really compile % this code on his/her own machine to ensure proper use of MEX. That can be done % from the MATLAB prompt with "mex decompToeplitzFastZohar.c". This C code does nothing % fancy, so if you can run MEX at all, this should work. See also 'help mex' and % 'mexext('all')'. % % MEX NOTE 2: The try-catch block is to hopefully fail MEX gracefully. If the mex % code is not compiled or in the wrong architecture, this will default to the next % best MATLAB implementation, which, depending on n, is inv() or vectorized Trench/Zohar. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [ld, Ti] = decompToeplitz(T_row, logdetMode, runMode) n = size(T_row, 2); % Dimension of the matrix % If runMode is specified, just run with that (for testing, etc.). Otherwise, do automated determination of runMode. if nargin < 3 % If runMode is not specified, do the automated determination of runMode, based on runtime tests tryMode = 0; % Always try C/MEX Zohar first. If MEX fails, use inv() or vectorized Zohar Trench in MATLAB. if n < 150 % The n=150 is a rough estimate based on tests on 64 and 32 bit linux systems catchMode = -1; % inv() is the best failure option when n < 150. else catchMode = 1; % n is >= 150, so vectorized Zohar is the best failure option. end else % If specified, force it. tryMode = runMode; catchMode = runMode; end % Invert T with the specified method. try [Ti, ld] = decompToeplitzMode(T_row, logdetMode, tryMode); catch [Ti, ld] = decompToeplitzMode(T_row, logdetMode, catchMode); end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Subfunction for try-catch block. This contains the actual Matlab algorithm, and calls the wrapper for the C/MEX function. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [Ti, ld] = decompToeplitzMode(T_row, logdetMode, runMode) switch (runMode) % Invert T to produce Ti, the inverse, and ld, the log determinant, by the specified method (or, if logdetMode is on, only the log determinant). case -1 T = convertToeplitz(T_row); % Convert our usual row vector notation into a full Toeplitz matrix. if not(logdetMode) Ti = inv(T); % Just call MATLAB inv(). elseif logdetMode Ti = 0; % Set an arbitrary (incorrect) value for Ti so that everything works smoothly. end ld = logdet(T); % Calculate the log determinant separately. case 0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Here is the fast C-MEX implementation of this inversion, using Zohar % invToeplitzFastZohar.c should be MEX-compiled with a command from the MATLAB % prompt of "mex invToeplitzFastZohar.c" This should just work. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if not(logdetMode) [Ti, ld] = decompToeplitzFast(T_row); % Call the MEX wrapper. elseif logdetMode ld = logdetToeplitzFastZohar(T_row); % Call the C/MEX function. Ti = 0; % Set an arbitrary (incorrect) value for Ti so that everything works smoothly. end case 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % A Matlab implementation of the Zohar algorithm. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% scale_factor = T_row(1, 1); % Save the scaling factor. T_row = T_row/scale_factor; % Normalize using the value on the central diagonal so that we can use the Zohar algorithm. n = size(T_row, 2)-1; r = transpose(T_row(1, 2:end)); % Intialize the value of r. In our code we have combined the variables r and rho from the paper, since they contain the same information. lambda = zeros(n, 1); % Initialize the vector of lambda values. % Initialize the algorithm. lambda(1, 1) = 1-r(1, 1)^2; % Obtain the first value of lambda. ghatold(1, 1) = -r(1, 1); % Obtain the first value of ghat, which is the same as g for the length-one case. ghat = ghatold; % Initialize ghat as ghatold. % Recursion to compute the values of gamma and lambda, and g and ghat. In our code we have combined the g and ghat variables, since they contain the same information. for i = 1:n-1 gamma(i, 1) = -(r(i+1, 1) + transpose(r(1:i, 1))*ghatold); % Compute gamma according to the Zohar algorithm. ghat = vertcat(gamma(i, 1)/lambda(i, 1), ghatold + gamma(i, 1)/lambda(i, 1)*ghatold(end:-1:1, 1)); % Compute the next value of ghat according to the Zohar algorithm. lambda(i+1, 1) = lambda(i, 1) - gamma(i, 1)^2/lambda(i, 1); % Compute lambda according to the Zohar algorithm. ghatold = ghat; % Save the new value of ghat in ghatold in preparation for the next recursive step. end if not(logdetMode) % If we're not in logdetMode, then construct and output the inverse, Ti. Ti(1, 1) = 1/lambda(n, 1); % Fill out the first element (upper left) of Ti. Ti(2:n+1, 1) = ghat(n:-1:1, 1)/lambda(n, 1); % Fill out the first column (left) of Ti. Ti(1, 2:n+1) = transpose(Ti(2:n+1, 1)); % Fill out the first row (top) of Ti by symmetry. Ti(n+1, 2:n+1) = Ti(1, n:-1:1); % Fill out the last row (bottom) of Ti by persymmetry and symmetry. Ti(2:n, n+1) = Ti(n:-1:2, 1); % Fill out the last column (right) of Ti by persymmetry and symmetry. % Fill out the rest of Ti using the equations from Zohar and symmetry. for j = 1:floor(n/2); Ti(j+1:n-j+1, j+1) = Ti(j:n-j, j) + 1/lambda(n, 1)*(ghat(n+1-j)*ghat(n+1-j:-1:1+j)-ghat(j:n-j)*ghat(j)); % Apply the Zohar equation to fill in one-fourth of the Ti output matrix. Ti(j+1, j+1:n-j+1) = Ti(j+1:n-j+1, j+1); % Use symmetry of Ti to fill out another quarter of the matrix. Ti(n+1-j:-1:j+1, n+2-(j+1)) = Ti(j+1:n-j+1, j+1); % Use persymmetry to fill out another quarter of the matrix. Ti(n+2-(j+1), n+2-(j+1):-1:n+2-(n-j+1)) = Ti(j+1, j+1:n-j+1); % Use persymmetry to fill out the final quarter of the matrix. end Ti = Ti/scale_factor; % Rescales Ti to produce the correct inverse. elseif logdetMode % If we are in logdetMode, then give an arbitrary filler value for Ti. Ti = 0; % Set an arbitrary (incorrect) value for Ti so that everything works smoothly. end ld = sum(log(lambda)) + log(scale_factor)*(n+1); % Calculates log(det(T)) for free (essentially, when the inverse is also computed) using the lambdas from our recursion. end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%PK CpJ (toeblitz-1.0/decomp/decompToeplitzFast.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % John P Cunningham, 2009 % modified by William Zhang, 2013 % % decompToeplitzFast() % % This function is simply a wrapper for the C-MEX % function decompToeplitzFastZohar.mexa64 (or .mexglx, etc), % which is just a compiled version of invToeplitzFastZohar.c, % which should also be in this folder. Please see % that code for details. % % This algorithm inverts a positive definite (symmetric) % Toeplitz matrix in O(n^2) time, which is considerably better % than the O(n^3) that inv() offers. This follows the Trench % algorithm implementation of Zohar 1969. See that paper for % all explanation, as the C code is just an implementation of % the algorithm of p599 of that paper. % % This function also computes the log determinant, which is % calculated essentially for free as part of the calculation of % the inverse. This is often useful in applications when one really % needs to represent the inverse of a Toeplitz matrix. % % This function should be called from within decompToeplitz.m, % which adds a try block so that it can default to a native % MATLAB inversion (either inv() or a vectorized version of % the Zohar/Trench algorithm, depending on the size of the matrix) % should the MEX interface not work. This will happen, for example, % if you move to a new architecture and do not compile for .mexmaci % or similar (see mexext('all') and help mex for some info on this). % % Inputs: % T_row the first row of a positive definite (symmetric) Toeplitz matrix T, which % does NOT need to be scaled to be 1 on the main diagonal. % % Outputs: % Ti the inverse of T % ld the log determinant of T, NOT Ti. % % % NOTE: This code is used to speed up the Toeplitz inversion as much % as possible. Accordingly, no error checking is done. The onus is % on the caller (which should be decompToeplitz.m) to pass the correct args. % % NOTE: Whenever possible, do not actually invert a matrix. This code is % written just in case you really need to do so. Otherwise, for example % if you just want to solve inv(T)*x for some vector x, you are better off % using a fast inversion method, like PCG with fast matrix multiplication (used in solveToeplitz.m), % which could be something like an FFT method for the Toeplitz matrix. To % learn about this, see Cunningham, Sahani, Shenoy (2008), ICML, "Fast Gaussian % process methods for point process intensity estimation." %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [Ti, ld] = decompToeplitzFast(T_row) [Ti, ld] = decompToeplitzFastZohar(T_row); end PK CpI -toeblitz-1.0/decomp/decompToeplitzFastZohar.c/*================================================================= * Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices * Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) *=================================================================*/ /*================================================================= * * decompToeplitzFastZohar.c completes the inversion of a symmetric * positive definite Toeplitz matrix. This * function is a subroutine of the decompToeplitz.m, which follows the * algorithm of Zohar 1969, a modification of the algorithm of W.F. Trench. * * The calling syntax is: * * [ Ti , logdetT ] = decompToeplitzFastZohar(T_row) * * T_row is the first row of the positive definite symmetric Toeplitz matrix T of dimension n+1 * (following the convention of p599 in the Zohar paper) * Ti is the inverse of T, same as inv(T) in MATLAB. * logdet is the log determinant of T, NOT of Ti. * * NOTE: This code is used to speed up the Toeplitz inversion as much * as possible. Accordingly, no error checking is done. The onus is * on the caller (which should be decompToeplitz.m) to pass the correct arguments. * * NOTE: This is superior to the Trench algorithm in Golub and * Van Loan's "Matrix Computations", since the Zohar version * also computes the log determinant for free, which is essential in the * application for which this algorithm was coded. * * John P Cunningham * 2009 * Modified by William Zhang, 2013 *=================================================================*/ /* Required Packages */ #include #include /* Input Arguments */ #define T_row_IN prhs[0] /* Output Arguments */ #define Ti_OUT plhs[0] #define LD_OUT plhs[1] void mexFunction( int nlhs, mxArray *plhs[], int nrhs, const mxArray*prhs[] ) { /* Declare variables */ double *T_row, *Ti, *logdet, *r, *lambda, *ghat, *ghatold, *gamma; unsigned int n; int i,j; /* Check for proper number of arguments*/ if (nrhs != 1) { mexErrMsgTxt("1 input arg required."); } else if (nlhs > 2) { mexErrMsgTxt("2 output args required."); } /* Check the dimensions of the arguments */ /* This expects T to be an n+1 by n+1 matrix */ n = mxGetN(T_row_IN)-1; /* Create a matrix for the return argument */ Ti_OUT = mxCreateDoubleMatrix(n+1, n+1, mxREAL); LD_OUT = mxCreateDoubleScalar(0); /* LD_OUT is a pointer to a 1x1 mxArray double initialized to 0. */ /* Assign pointers to some input and output parameters */ Ti = mxGetPr(Ti_OUT); logdet = mxGetPr(LD_OUT); T_row = mxGetPr(T_row_IN); /*Allocate space on heap for various variables.*/ r = (double*) mxMalloc(n*sizeof(double)); if (r == NULL) { mexErrMsgTxt("Inadequate space on heap for r."); } lambda = (double*) mxMalloc(n*sizeof(double)); if (lambda == NULL) { mexErrMsgTxt("Inadequate space on heap for lambda."); } ghat = (double*) mxMalloc(n*sizeof(double)); if (ghat == NULL) { mexErrMsgTxt("Inadequate space on heap for ghat."); } ghatold = (double*) mxMalloc(n*sizeof(double)); if (ghatold == NULL) { mexErrMsgTxt("Inadequate space on heap for ghatold."); } gamma = (double*) mxMalloc(n*sizeof(double)); if (gamma == NULL) { mexErrMsgTxt("Inadequate space on heap for gamma."); } /* Define r, the normalized row from T_row(1,2) to T_row(1,n+1) */ for (i = 0; i < n ; i++) { r[i] = T_row[0*(n+1) + (i+1)]/T_row[0*(n+1) + 0]; } /* Initialize the values of lambda and g for the algorithm */ lambda[0] = 1 - pow(r[0],2); ghat[0] = -r[0]; /* Recursion to build g and lambda */ for (i=0 ; i < n-1 ; i++) { /* Calculate gamma */ gamma[i] = -r[i+1]; /* Note that ghat, g_i, etc are i+1 elements long */ for (j=0 ; j < i+1 ; j++) { gamma[i] -= r[j]*ghat[j]; } /* Calculate next ghat */ /* First store ghatold, which is i+1 elts long */ for (j=0 ; j < i+1 ; j++) { ghatold[j] = ghat[j]; } /* The first element of the new ghat is derived from gamma and lambda. */ ghat[0] = gamma[i]/lambda[i]; /* The rest of the new ghat is formed from the old ghat, ghatold, as described on page 599 of the Zohar paper. */ for (j=1 ; j < i+2 ; j++) { ghat[j] = ghatold[j-1] + gamma[i]/lambda[i]*ghatold[i+1-j]; } /* Calculate lambda */ lambda[i+1] = lambda[i] - pow(gamma[i],2)/lambda[i]; } /* There are n lambdas, n g values, and n-1 gammas */ /* Note that n is not the dimension of the matrix, but one less. */ /* This corresponds with Zohar notation. */ /* Evaluate the matrix Ti (B_{n+1} in Zohar) */ /* NOTE ON MEX MATRIX INDEXING */ /* indexing is always (colidx * colLen + rowidx) */ /* Assign first upper left element */ Ti[ 0*(n+1) + 0 ] = 1/lambda[n-1]; /* Assign the first row and column*/ for (i = 1; i < n+1; i++) { /* First row */ Ti[ 0*(n+1) + i] = ghat[ n-1-(i-1) ]/lambda[ n-1 ]; /* First column */ Ti[ i*(n+1) + 0] = ghat[ n-1-(i-1) ]/lambda[ n-1 ]; } /* Fill in last row and column by persymmetry and symmetry with first row and column.*/ for (i = 1; i < n ; i++) { /* Last row */ Ti[ n*(n+1) + i ] = ghat[ n-1-(n-i-1) ]/lambda[ n-1 ]; /* Last column */ Ti[ i*(n+1) + n ] = ghat[ n-1-(n-i-1) ]/lambda[ n-1 ]; } /* Assign the last (bottom right) element by persymmetry */ Ti[ n*(n+1) + n ] = Ti[ 0*(n+1) + 0]; /* Fill in the interior of Ti_OUT */ for (i = 0; i < n/2 ; i++) { for (j = i; j < n-i-1; j++) { /* Calculate the value using equations on page 599 of Zohar. */ Ti[ (j+1)*(n+1) + (i+1) ] = (Ti[ j*(n+1) + i ] + (1/lambda[n-1])*(ghat[n-1-i]*ghat[n-1-j] - ghat[n-1-(n-1-i)]*ghat[n-1-(n-1-j)])); /* Use symmetry */ Ti[ (i+1)*(n+1) + (j+1) ] = Ti[ ((j+1)*(n+1)) + (i+1) ]; /* Use persymmetry: Recall there are n+1 elements, so 0<->n, 1<->n-1, and so on */ Ti[ (n-(i+1))*(n+1) + (n-(j+1)) ] = Ti[ ((j+1)*(n+1)) + (i+1) ]; Ti[ (n-(j+1))*(n+1) + (n-(i+1)) ] = Ti[ ((j+1)*(n+1)) + (i+1) ]; } } /* Normalize the entire matrix by T(1,1) so it is the properly scaled inverse */ for (i = 0; i < n+1 ; i++) { for (j = 0; j < n+1 ; j++) { Ti[ j*(n+1) + i ] = Ti[ j*(n+1) + i ]/T_row[ 0*(n+1) + 0]; } } /* Calculate the log determinant for free (essentially) by summing the logs of the lambdas. */ logdet[0] = 0; for (i=0 ; i < n; i++) { logdet[0] += log(lambda[i]); } /* Renormalize based on T_row(1,1) */ logdet[0] += (n+1)*log(T_row[ 0*(n+1) + 0 ]); /* Free allocated arrays */ mxFree(gamma); mxFree(ghatold); mxFree(ghat); mxFree(lambda); mxFree(r); return; }PK C]+5+5/toeblitz-1.0/decomp/decompToeplitzFastZohar.mexELF4L)4 ($!HH$$QtdRtdttGNUVU`~ڱz6X     ` |CEqX4K _+ "Xt4 , , ?@F    __gmon_start___init_fini__cxa_finalize_Jv_RegisterClassesmexFunctionmexErrMsgTxtmxGetNmxCreateDoubleMatrixmxCreateDoubleScalarmxGetPrmxMalloclogmxFreeliboctinterp.soliboctave.solibcruft.soliblapack.so.3gflibblas.so.3gflibfftw3.so.3libfftw3f.so.3libreadline.so.6libncurses.so.5libdl.so.2libhdf5.so.6libz.so.1libgfortran.so.3libstdc++.so.6libm.so.6libgcc_s.so.1libc.so.6_edata__bss_start_endGLIBC_2.0GLIBC_2.1.3n ii si (           $  US[Lt.=X[ hhhhh h($h0(h8p,h@`0hHPUVSj8u]t4$h<)9s <<9rƃ8[^]US.tt $Ѓ[]Ë$ÐUWVSu à}}x$g$D$EȃEEȉD$$]$F$tEԋF$fE$YtEE$"E2U$ E/M $-}<$E*E$E'Mt11}uܐ4׉19EwUM}uȃ}}ثUEU1ҍEЋ}MljE܋EM1Mf ƃ9~؋M19~M܃1MU1M܋MĉUU2U \9EuȋM܋U؋}̋EE;ULMU}ыM2UvF}Ⱥu}؋}uE)֍u77U9Mu֋u܃}vS}}ȉu܉}؍u܃}؍֋uUU6u9M}7։w͋u܋EUԋMEM}ȋUȋMȋEȃ}}|UȉMفE1UEExMu؍U+Ũ9ЉUu؍H}4֋U؉uu̍u܋u11&M$ݝhE9݅hw֋u܋t$ݝhE1҉UM}Em݅h<$E$4$U$M $ļ[^_]Ã}$y}EEEث $@$h$$n$[UVSZt&Ћu[^]US[ Y[1 input arg required.2 output args required.Inadequate space on heap for r.Inadequate space on heap for lambda.Inadequate space on heap for ghat.Inadequate space on heap for ghatold.Inadequate space on heap for gamma.? ,7DN_nx  o  PL, oooo"2BRbr( GCC: (Ubuntu 4.4.3-4ubuntu5.1) 4.4.3@FmexFunction@  int!95s&ni2 ]1@  1(, A1X 1( K1 W4* Ti4S 4u r4 P4uF44 4u n5Df i6(y j6(==6% $ > $ > : ; 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N, Uj ~"@F  crtstuff.c__CTOR_LIST____DTOR_LIST____JCR_LIST____do_global_dtors_auxcompleted.7021dtor_idx.7023frame_dummy__CTOR_END____FRAME_END____JCR_END____do_global_ctors_auxdecompToeplitzFastZohar.c__i686.get_pc_thunk.bx_GLOBAL_OFFSET_TABLE__DYNAMIC__DTOR_END____dso_handle_finimexErrMsgTxt__bss_start_end__gmon_start___edatamxCreateDoubleMatrix_Jv_RegisterClassesmxMalloc__cxa_finalize@@GLIBC_2.1.3mexFunctionmxGetPr_initlog@@GLIBC_2.0mxGetNmxCreateDoubleScalarmxFreePK CEe2#2#2toeblitz-1.0/decomp/decompToeplitzFastZohar.mexa64ELF>@@8@    HH H $$PtdQtdGNU=qbv(>6 @  5 8L  l"?S 3 __gmon_start____cxa_finalize_Jv_RegisterClassesmexFunctionmexErrMsgTxtmxGetNmxCreateDoubleMatrix_700mxCreateDoubleScalarmxGetPrmxMalloclogmxFreelibmx.solibmex.solibmat.solibm.so.6libstdc++.so.6libpthread.so.0libc.so.6decompToeplitzFastZohar.mexa64MEXGLIBC_2.2.5$Q  ui ui @ @ H P X x             HJH5 % @% h% h% h% h% h% h% h% hp% h`HHM HtHÐU= HATSubH=0 t H= ZH L% H} L)HHH9s DHH] AHR H9r> [A\fH= UHtH HtH= @ÐUHAWAVAUATSHHI΃tH=~ H=I>nAAUUƉHLcfWVI$H;*HI<$HhI>H`EHHEH5IHpHu H=O:H}HEHu H=QH}IIHu H=YH}HEHXHu H=\H}HEHu H=f}}LpH`Hcȃ^A9wHpY 7\LUA  .fWAEEDHcL4HMLLCDfWDHE~8LA AY \HH9uLI<H<HH9uLuA^HUP~-LA^HcAYXADHuf(YA^\HMBHD;MLuH]DmELUIHE^AvEDEHuD)I Ή^A^ӃA9w̓}vN}AuLEI D :A^BAA^Ӄ9wʋEEHHËE艅|@AUUDىMEUDxEDUEEEED؃ADH AEDMf(LUA^ DUA)ACYAA$CY\YBXEÉL L ÉH H Eu;UyEEEDmUU9|vR‰ЋM+MȉM9sӉM<΋MM$DBEDUGD֋}}uAx+MEtIL`+AHA^A9w߃A9vHhH}tKLuDuIELmA$EHcADXEA$A9wLuELhAEL`AnEI*YXEHhH}HX LH}HpHĈ[A\A]A^A_HpY \HM  fWAE}ALXLuH]H]DmLpFUHSHH HtH HHHuH[ÐHH1 input arg required.2 output args required.Inadequate space on heap for r.Inadequate space on heap for lambda.Inadequate space on heap for ghat.Inadequate space on heap for ghatold.Inadequate space on heap for gamma.?; 0zRx ,AC S A @  8 oH  ` `` ooooodoH fvGCC: (GNU) 4.4.6 20120305 (Red Hat 4.4.6-4)GCC: (GNU) 4.4.7 20120313 (Red Hat 4.4.7-3).symtab.strtab.shstrtab.note.gnu.build-id.gnu.hash.dynsym.dynstr.gnu.version.gnu.version_d.gnu.version_r.rela.dyn.rela.plt.init.text.fini.rodata.eh_frame_hdr.eh_frame.ctors.dtors.jcr.data.rel.ro.dynamic.got.got.plt.bss.comment$.o(8 h@HHHoddUo8do@s`}`` 88PPL ( (8 8@ @H HH H` `` 0XX2 @!Hd ` 8 P   ( 8 @ H H `    *( 88 E [ j x   8  P ` @ 0   " )H 2 88@G V jn" call_gmon_startcrtstuff.c__CTOR_LIST____DTOR_LIST____JCR_LIST____do_global_dtors_auxcompleted.6349dtor_idx.6351frame_dummy__CTOR_END____FRAME_END____JCR_END____do_global_ctors_auxdecompToeplitzFastZohar.c_fini_GLOBAL_OFFSET_TABLE___dso_handle__DTOR_END____bss_start_end_edata_DYNAMIC_initmxGetPrmxGetN__gmon_start___Jv_RegisterClassesMEXmxCreateDoubleScalar__cxa_finalize@@GLIBC_2.2.5mxMallocmxFreemexFunctionmexErrMsgTxtmxCreateDoubleMatrix_700log@@GLIBC_2.2.5PK C^!!5toeblitz-1.0/decomp/decompToeplitzFastZohar.mexmaci64x__TEXT__text__TEXT.__stubs__TEXT 0 __stub_helper__TEXT ` __const__TEXTP P__cstring__TEXTpp__unwind_info__TEXT\P\__eh_frame__TEXTP__DATA__dyld__DATA__la_symbol_ptr__DATA@H__LINKEDIT  H 8! P  99e3 MA$  0@rpath/libmx.dylib 0@rpath/libmex.dylib 0@rpath/libmat.dylib 84/usr/lib/libstdc++.6.dylib 8/usr/lib/libgcc_s.1.dylib 8{/usr/lib/libSystem.B.dylib&@ )H ASLgAS%_ %^ UHAWAVAUATSHHIt H= | H=H;IL}1DDIfIFI>II~HhH;HEAGHPHHEHHEHu H=5rH}cHEHu H=7TH}EHEHu H=A6H}'HEHu H=KH} HEHu H=UHPt,HEȍ@1@HUD^HUHH9uHEY \HM fWHEHEȃHH@HFHHMHMHIHM1ɺL]DHu fWHuL~uHH}LEf.AY\LMALIHHuۅ~7HH}LEfDfDAIHHuHuLHu^LHu|`HHLELMDfDLUALLUA^LII CY AXA ILIIHuHuLH}TY^\ HHHuHtHHHEȍHHMLHU^A$rOID$HMȍQLEH}@HuB^B^ljAHMHHHRuHPw0HEȍ@HPA$AHEȅM1 HEȍHHUHRt@HLEDfDH}B^ACB^ljAHuHHHuHEȍ@HPA$AăaHHHHGH`HMȍAHEHXHEQAI|1HpHMHufHpH<0D|IALUIHMDfDf(LmL}C^ L}CACYAY\YACXACACC EC C EC HuLUHHIIIsHpHuHXHu|HHHuHHuH9`Hu9fHMȉʉAHu^AHuuH9uHhHHPusfEHEHEȉH*YXMHhH}H}H}xH}oH}HĈ[A\A]A^A_]UHEȉfELufDANXEHhIHOE%L%N%P%R%T%V%X%ZLLL|L pL dLXLLL@?1 input arg required.2 output args required.Inadequate space on heap for r.Inadequate space on heap for lambda.Inadequate space on heap for ghat.Inadequate space on heap for ghatold.Inadequate space on heap for gamma.44 4  XzRx 4 __  ,8D (08@H  .;@Nh~   dyld_stub_binding_helper__dyld_func_lookup_mexFunction_log_mexErrMsgTxt_mxCreateDoubleMatrix_700_mxCreateDoubleScalar_mxFree_mxGetN_mxGetPr_mxMallocPK CkM,,2toeblitz-1.0/decomp/decompToeplitzFastZohar.mexw64MZ@ !L!This program cannot be run in DOS mode. $`wwwwww܃wywvww݃wwwwwRichwPEd R"  #cJ@7]3d`P p 0.text `.rdata0@@.data`@$@.pdata P&@@.rsrc`(@@.reloc p*@BHSUVWATAWHLhLp)xD)@IHD)HAt H ! ~ H !<HM-E3DhL$Au֋Ή$XHfWHHGHOHHD$8HMH$EL$IHIHD$`HH$Hu H !IuLHu H IYLHD$XHu H ^I8H$Hu H @ =ILH$Hu H E3E3A|oAEHHMH+DD LAH I^ED I^ML A^ED I^ML uE;s*LAJ L+A+ADHH^EAuD 4fA(^ =AUf(HD$0A\ȉT$(A $DfAWA`H$H$I)$IHLH+I+MHD$@L+AHL$PL+HD$HfffHD9E3BT(3fAWAAAIHKH+DDHfDD H IYA\L YI\D YA\L YI\uE;},MAH L+A+BHHYA\uE3E3A|[AAIHNH+DDLHD H IHAHD HAHD HAHD HAuE;}!MAJ L+A+JHHHAuEQHDHD$ D$`A^>AD;AHD$PLIOHT8AAM+DDHHD$ H H IA^>YB0AXDAA^ >YJ(AXLIA^>YB AXDAA^ >YJAXLIuE;}GI HD$0LH+D$ EM+HTE+HHIA^>YBBXD AuA4>fA( HL$PHT$@^HD$0D$`HD9HHL$H\tH\$8$XHl$`($L$T$(AD($D($f(EMcIA^F!BA$`CD-E++FKTD$`GDmLS$XH ffBA(I H C^DABJ0C^L B(C0C^DABJ(C^L B A0C^DABJ C^LB BEHC^DABJC^L SHl$`L|$XD$XD;sMILˋH+AI NjA+DIHC^DA@IC^L IuIAAE*BADƍHDPCD-Eʼn$XA+AEE+IWD$XD HHD$ BA@H C^DJA*C^L BC2C^DBJC^L BA@C^DBA@AC^DBJC^L BA2EHC^DJHl$`L|$XHL$ E;sIDMAAEAC AIC^DA@C^DE;rHAME3E3LL$PAHA艄$DFKTID$XHT$0L|$ AAMA+LL$HɉL$(D; A+ƃL|$0DQƉD$|BD$lHEAHD$8FDT$tABT0IGHD$@AF$`+ȍAQƉ$֍BT$p$AƉ$ȉ$AD$xA@D$`D$hAA+I HD$8HL$HHL$@@QD$xDT$hDL$|AAYYEf(\DD+EC^LAIYf(XBEŋD+ABHHA@A+BL(HӋ$`HHD$@AIC^LHD$8@AYYE$\Yf(XBHHA@A+BL(HӋ$`HHD$@AIC^LPHD$8AYYEB\YXË$‹ABHHA@A+f(σBL(HӋ$`HHD$@AIC^LPHD$8@AYYE$\YXË$‹ABHHA@D$`DD$lA+BL(HӋT$pHHD$8HL$@H H D$`HD$8HL$@;|$tL|$XL$(HT$0D$XLL$PLT$H;l$(O Ɖ$`GDFT0IDEH+D$HM|AHD$ f(IIC^L$XYAYAHT$0\YBXDBË$`+BL(JHA+A+BL(JDH;nL|$XHl$ D$XLL$PDIHHAD$XLL$PHT$0Hl$ D;$Hl$`($E3ۅfD3CD-AABTD+ЍFE+F4D<B^EB ^EӍ2^EAI^Eu;s"΋A+^EHuAD;?H$L$EL$PHt*L$IHIXuL|$XEH$fH*YX9H$,I$IH$HA_A\_^][%%%%r%d%V%@%%@SH HHH H HuC#H#H H 3H [HHXHhHxL` AUAVAWH 3ML8#DeH%0HXH;t3HuAtcH LHH MLHHI;rZH9}tH9EtHMTHHEH V8H AH(L;uL;tLLIEGHH=EH=3eH%0HXH;t3Hut ;>H|H ekuHCH 4Eu HH?H9=Pt!H GtMĺI-H\$@Hl$HH|$PLd$XH A_A^A]HHXHpHxATH0ILXu9u 3ۉXtu7HHtЋ؉D$ LƋI0؉D$ LƋI؉D$ u5u1L3IL3IL9Mt L3IAӅtu7LƋI#ˋىL$ tHHtLƋIЋ؉D$ 3ۉ\$ H\$@Ht$HH|$PH0A\H\$Ht$WH IHuoLNjHH\$0Ht$8H _H M@SH HH `B HD$8Hu H ~H 2 HD$8H  HD$@H HLD$@HT$8HHL$8 HHL$@ HWHH [H(GHH(H\$WH H H= HHtHH;rH\$0H _H\$WH H H=x HHtHH;rH\$0H _HMZf9t3HcH6F6V6d6|66666606555555x5?@Inadequate space on heap for gamma.Inadequate space on heap for ghatold.Inadequate space on heap for ghat.Inadequate space on heap for lambda.Inadequate space on heap for r.2 output args required.1 input arg required.!3!2!2!2!h2! 2!x2!"*3# p`P020  t T 42t d 4R%"i#&i#}"o# 'd42 p20%$}$:'  4 2p2P B%%%`'%B  4 2p@550055046P0`470v7`7J7:7 7766666$65>6F6V6d6|66666606555555x5mxFree5mxMallocmxGetPrQmxCreateDoubleScalarOmxCreateDoubleMatrix_700mxGetNlibmx.dll1mexErrMsgTxtlibmex.dllpowlogMSVCR100.dll_malloc_crt_initterm_initterm_ecfree_encoded_null_amsg_exit__C_specific_handler__CppXcptFilter@__clean_type_info_names_internal[_unlockH__dllonexit_lock_onexitEncodePointerDecodePointerSleepDisableThreadLibraryCallsQueryPerformanceCounterGetTickCountGetCurrentThreadIdGetCurrentProcessIdGetSystemTimeAsFileTimeKERNEL32.dll R77777decompToeplitzFastZohar.mexw64mexFunctionu2-+] f32222222fx2$3U",3X"#H3##3#$3$$3$$3$%3%%3&#&3$&&3&'3 ':'3:'U'3`''30 HX`Z PAPADDINGXXPADDINGPADDINGXXPADDINGPADDINGXXPADDINGPADDINGXXPADDINGPADDINGXXPAD0 0PK C7qtoeblitz-1.0/decomp/logdet.m% log(det(A)) where A is positive-definite. % This is faster and more stable than using log(det(A)). % Written by Tom Minka % (c) Microsoft Corporation. All rights reserved. % Additional comments added by William Zhang, 2013. function [y] = logdet(A) U = chol(A); % Standard Cholesky decomposition A = U'*U y = 2*sum(log(diag(U))); % The product of the diagonal % of any triangular matrix is the determinant. endPK CFF-toeblitz-1.0/decomp/logdetToeplitzFastZohar.c/*================================================================= * Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices * Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) *=================================================================*/ /*================================================================= * * Adapted from decompToeplitzFastZohar.c. * logdetToeplitzFastZohar uses the Zohar algorithm to determine * the log determinant of a positive definite Toeplitz matrix, without * performing the computationally costly task of constructing the inverse. This * function is a subroutine of the logdetToeplitz.m, which follows the * algorithm of Zohar 1969, a modification of the algorithm of W.F. Trench. * * The calling syntax is: * * [ logdetT ] = logdetToeplitzFastZohar(T_row) * * T_row is the first row of the positive definite symmetric Toeplitz matrix T of dimension n+1 * (following the convention of p599 in the Zohar paper) * logdet is the log determinant of T. * * NOTE: This code is used to speed up the Toeplitz inversion as much * as possible. Accordingly, no error checking is done. The onus is * on the caller (which should be decompToeplitz.m) to pass the correct arguments. * * NOTE: This is superior to the Trench algorithm in Golub and * Van Loan's "Matrix Computations", since the Zohar version * also computes the log determinant for free, which is essential in the * application for which this algorithm was coded. * * John P Cunningham * 2009 * Modified by William Zhang, 2013 *=================================================================*/ /* Required Packages */ #include #include /* Input Arguments */ #define T_row_IN prhs[0] /* Output Arguments */ #define LD_OUT plhs[0] void mexFunction( int nlhs, mxArray *plhs[], int nrhs, const mxArray*prhs[] ) { /* Declare variables */ double *T_row, *logdet, *r, *lambda, *ghat, *ghatold, *gamma; unsigned int n; int i,j; /* Check for proper number of arguments*/ if (nrhs != 1) { mexErrMsgTxt("1 input arg required."); } else if (nlhs > 2) { mexErrMsgTxt("2 output args required."); } /* Check the dimensions of the arguments */ /* This expects T to be an n+1 by n+1 matrix */ n = mxGetN(T_row_IN)-1; /* Create a matrix for the return argument */ LD_OUT = mxCreateDoubleScalar(0); /* LD_OUT is a pointer to a 1x1 mxArray double initialized to 0. */ /* Assign pointers to some input and output parameters */ logdet = mxGetPr(LD_OUT); T_row = mxGetPr(T_row_IN); /*Allocate space on heap for various variables.*/ r = (double*) mxMalloc(n*sizeof(double)); if (r == NULL) { mexErrMsgTxt("Inadequate space on heap for r."); } lambda = (double*) mxMalloc(n*sizeof(double)); if (lambda == NULL) { mexErrMsgTxt("Inadequate space on heap for lambda."); } ghat = (double*) mxMalloc(n*sizeof(double)); if (ghat == NULL) { mexErrMsgTxt("Inadequate space on heap for ghat."); } ghatold = (double*) mxMalloc(n*sizeof(double)); if (ghatold == NULL) { mexErrMsgTxt("Inadequate space on heap for ghatold."); } gamma = (double*) mxMalloc(n*sizeof(double)); if (gamma == NULL) { mexErrMsgTxt("Inadequate space on heap for gamma."); } /* Define r, the normalized row from T_row(1,2) to T_row(1,n+1) */ for (i = 0; i < n ; i++) { r[i] = T_row[0*(n+1) + (i+1)]/T_row[0*(n+1) + 0]; } /* Initialize the values of lambda and g for the algorithm */ lambda[0] = 1 - pow(r[0],2); ghat[0] = -r[0]; /* Recursion to build g and lambda */ for (i=0 ; i < n-1 ; i++) { /* Calculate gamma */ gamma[i] = -r[i+1]; /* Note that ghat, g_i, etc are i+1 elements long */ for (j=0 ; j < i+1 ; j++) { gamma[i] -= r[j]*ghat[j]; } /* Calculate next ghat */ /* First store ghatold, which is i+1 elts long */ for (j=0 ; j < i+1 ; j++) { ghatold[j] = ghat[j]; } /* The first element of the new ghat is derived from gamma and lambda. */ ghat[0] = gamma[i]/lambda[i]; /* The rest of the new ghat is formed from the old ghat, ghatold, as described on page 599 of the Zohar paper. */ for (j=1 ; j < i+2 ; j++) { ghat[j] = ghatold[j-1] + gamma[i]/lambda[i]*ghatold[i+1-j]; } /* Calculate lambda */ lambda[i+1] = lambda[i] - pow(gamma[i],2)/lambda[i]; } /* There are n lambdas, n g values, and n-1 gammas */ /* Note that n is not the dimension of the matrix, but one less. */ /* This corresponds with Zohar notation. */ /* NOTE ON MEX MATRIX INDEXING */ /* indexing is always (colidx * colLen + rowidx) */ /* Calculate the log determinant for free (essentially) by summing the logs of the lambdas. */ logdet[0] = 0; for (i=0 ; i < n; i++) { logdet[0] += log(lambda[i]); } /* Renormalize based on T_row(1,1) */ logdet[0] += (n+1)*log(T_row[ 0*(n+1) + 0 ]); /* Free allocated arrays */ mxFree(gamma); mxFree(ghatold); mxFree(ghat); mxFree(lambda); mxFree(r); return; } PK CR##/toeblitz-1.0/decomp/logdetToeplitzFastZohar.mexELF44 ($!T T HH$$QtdRtdttGNU\y H|20Ax[`'nt     ` |CEqX4K + |"tX_0 {( ( ?A 0 H  __gmon_start___init_fini__cxa_finalize_Jv_RegisterClassesmexFunctionmexErrMsgTxtmxGetNmxCreateDoubleScalarmxGetPrmxMalloclogmxFreeliboctinterp.soliboctave.solibcruft.soliblapack.so.3gflibblas.so.3gflibfftw3.so.3libfftw3f.so.3libreadline.so.6libncurses.so.5libdl.so.2libhdf5.so.6libz.so.1libgfortran.so.3libstdc++.so.6libm.so.6libgcc_s.so.1libc.so.6_edata__bss_start_endGLIBC_2.0GLIBC_2.1.3Y ii qsi $           US[øt.)X[ hhhhh h($h0(h8p,h@`UVS4u]t0$l8)9s 889rƃ4[^]US.ktt $Ѓ[]Ë$ÐUWVSlu#}tp$$E$EU $E$}<$EYEs<$FEs<$3t<$!Et<$EUM11}ĉu܍v4׉19EwEċUԋMuȃMثX}1E}ȍv}ЋEЋM}܋}}E1t& ƃ9~؋M19~E܃EM0E1UЉUU2U \;EuȋM܋}̃E9EUMLE}t5؉u܋u11Uԃ$]M9Ew܋u܋}$]E1҉UUEEmE$M $4$}ԉ<$Eĉ$l[^_]Ã}$}UċMEثXPE4$$z$z$y $tyUVSt&Ћu[^]US[àY[1 input arg required.2 output args required.Inadequate space on heap for r.Inadequate space on heap for lambda.Inadequate space on heap for ghat.Inadequate space on heap for ghatold.Inadequate space on heap for gamma.?"/9JYcq 0 H op  H ooofov$ GCC: (Ubuntu 4.4.3-4ubuntu5.1) 4.4.3AmexFunction}  int25lgbV1 ]1(,:1]X 1(D1iP4z  4zu r4z I4zuT ?4z 4z y4zu n5D i6(/ j6(n c = o u= 6% $ > $ > : ; $ > .? : ; ' @: ; I4: ; I 4: ; I 4: ; I 4: ; I 4: ; I  I&I! /home/cunni/toeplitz/wemmick/toeplitz/decomp/usr/include/octave-3.2.3/octavelogdetToeplitzFastZohar.cmex.h1n *@8@@Y-=k-=k-/k-=k-=m>,5jbNbic&(dZH dz?9[9[^dfd*f;Y-/:>Lr^x8$;.gs,Gh.444| AAB Inlhslogdetnrhsunsigned charshort unsigned intmxArrayplhsghatprhslambdaT_rowmexFunctionlong long unsigned intgammaghatoldlong long intGNU C 4.4.3/home/cunni/toeplitz/wemmick/toeplitz/decomp/logdetToeplitzFastZohar.cshort intlong doublettAuuAu Au !!Au! !VuVuVAuQ/VP+RR2W2?R\P.symtab.strtab.shstrtab.note.gnu.build-id.gnu.hash.dynsym.dynstr.gnu.version.gnu.version_r.rel.dyn.rel.plt.init.text.fini.rodata.eh_frame.ctors.dtors.jcr.dynamic.got.got.plt.data.bss.comment.debug_aranges.debug_pubnames.debug_info.debug_abbrev.debug_line.debug_frame.debug_str.debug_loc$2X.opp<8 @Hoff"Uo@d  m H v000q``|HH H d d P P H 0$ $( (0(%M m"%4*0$5"@@#4 !pf 0 `  H  d P $ (   (5 K( Z, h tP    $ H  !.( :0 ? N( U ir"A 0 crtstuff.c__CTOR_LIST____DTOR_LIST____JCR_LIST____do_global_dtors_auxcompleted.7021dtor_idx.7023frame_dummy__CTOR_END____FRAME_END____JCR_END____do_global_ctors_auxlogdetToeplitzFastZohar.c__i686.get_pc_thunk.bx_GLOBAL_OFFSET_TABLE__DYNAMIC__DTOR_END____dso_handle_finimexErrMsgTxt__bss_start_end__gmon_start___edata_Jv_RegisterClassesmxMalloc__cxa_finalize@@GLIBC_2.1.3mexFunctionmxGetPr_initlog@@GLIBC_2.0mxGetNmxCreateDoubleScalarmxFreePK CO!2toeblitz-1.0/decomp/logdetToeplitzFastZohar.mexa64ELF>@@8@      $$Ptd QtdGNUu,L B @  5 hL  S"p}?y3 \__gmon_start____cxa_finalize_Jv_RegisterClassesmexFunctionmexErrMsgTxtmxGetNmxCreateDoubleScalarmxGetPrmxMalloclogmxFreelibmx.solibmex.solibmat.solibm.so.6libstdc++.so.6libpthread.so.0libc.so.6logdetToeplitzFastZohar.mexa64MEXGLIBC_2.2.5 Q ui ui      ( H P X ` h p  x    H:%H52 %4 @%2 h%* h%" h% h% h% h% h% hpHH} HtHÐU= HATSubH=` t H=? jH# L% H L)HHH9s DHH AHz H9rf [A\fH= UHtH HtH= @ÐUHAWAVAUATSHXIH˃tH=~ H=H;EEfWI$HSHEH;GHEEHHEHpIIHu H=yH}PHEHu H=[H}2HHEHu H=:H}IIHu H=H}HEHu H=}}HUHcȃ^A9wAY \HEA fWHEEEDHcLHMLL CfWDBE~8LA Y \HH9uLH<I|HH9uL]A^HEB~.LA^HcAYAXDDHuf(YA^\HEBHD;EHEH}tELuDuLmIA$EHcADjXEA$A9wLuHEEHE9EH*YXEHEH}LH}H}LHX[A\A]A^A_AY \HEA fWHEEAHEHLuDuLmIUHSHH HtH HHHuH[ÐHOH1 input arg required.2 output args required.Inadequate space on heap for r.Inadequate space on heap for lambda.Inadequate space on heap for ghat.Inadequate space on heap for ghatold.Inadequate space on heap for gamma.?;0zRx ,AC P) A    X o0  0 (` oPoooo4o &6FVfvGCC: (GNU) 4.4.6 20120305 (Red Hat 4.4.6-4)GCC: (GNU) 4.4.7 20120313 (Red Hat 4.4.7-3).symtab.strtab.shstrtab.note.gnu.build-id.gnu.hash.dynsym.dynstr.gnu.version.gnu.version_d.gnu.version_r.rela.dyn.rela.plt.init.text.fini.rodata.eh_frame_hdr.eh_frame.ctors.dtors.jcr.data.rel.ro.dynamic.got.got.plt.bss.comment$.o(8 P@00Ho44UoPP8do@s`}(( X X p p   L         0 0X 0X 2 04P (    X p         0    * 8 E [ j x 0     X 0     " ) 2 8@G V jn" \call_gmon_startcrtstuff.c__CTOR_LIST____DTOR_LIST____JCR_LIST____do_global_dtors_auxcompleted.6349dtor_idx.6351frame_dummy__CTOR_END____FRAME_END____JCR_END____do_global_ctors_auxlogdetToeplitzFastZohar.c_fini_GLOBAL_OFFSET_TABLE___dso_handle__DTOR_END____bss_start_end_edata_DYNAMIC_initmxGetPrmxGetN__gmon_start___Jv_RegisterClassesMEXmxCreateDoubleScalar__cxa_finalize@@GLIBC_2.2.5mxMallocmxFreemexFunctionmexErrMsgTxtlog@@GLIBC_2.2.5PK C=F(!!5toeblitz-1.0/decomp/logdetToeplitzFastZohar.mexmaci64x__TEXT__text__TEXT  __stubs__TEXT * __stub_helper__TEXT T __const__TEXTP P__cstring__TEXTpp__unwind_info__TEXT\P\__eh_frame__TEXTP__DATA__dyld__DATA__la_symbol_ptr__DATA8H__LINKEDIT  @ ! P  k'6پI$  0@rpath/libmx.dylib 0@rpath/libmex.dylib 0@rpath/libmat.dylib 84/usr/lib/libstdc++.6.dylib 8/usr/lib/libgcc_s.1.dylib 8{/usr/lib/libSystem.B.dylib&8 )@ ASLAS%%UHAWAVAUATSHHHIt H= | H=%H;jIL}fNIHUHEH;IHEAGHEHH3HEHu H=$HIMu H=HHEHu H=HIMu H=HHHu H=HEt0HE@1HUD^HUHH9uHEY \AfWHEHE[HH@HFHHMHMHIHM1ɺ|HfH} fWL~fHLELMAAY\LIIHu߅~(HMLMАA AIIHuLA^LH}|CHIMLUfLA^LMI CY AX A IIIHuALTY^\A HHH}H|HHHEHHEujfEHEHEH*YXMHEH}LuH}lLdH}HH[A\A]A^A_]MHEAAfEMDAEEXEHEIIZE%>%@%B%D%F%H%JL L L LLLL?1 input arg required.2 output args required.Inadequate space on heap for r.Inadequate space on heap for lambda.Inadequate space on heap for ghat.Inadequate space on heap for ghatold.Inadequate space on heap for gamma. 44 4  XzRx 4`  __ (4@ (08@  $ .0 ;@Ndlt}   dyld_stub_binding_helper__dyld_func_lookup_mexFunction_log_mexErrMsgTxt_mxCreateDoubleScalar_mxFree_mxGetN_mxGetPr_mxMallocPK Cz &$$2toeblitz-1.0/decomp/logdetToeplitzFastZohar.mexw64MZ@ !L!This program cannot be run in DOS mode. $vvvv݃v޹vwv܃vvvRichvPEd R"  4p@@']#dP@` .text" `.rdata @@.data`0@.pdata@@@.rsrcP @@.reloc `"@BHHXHhHp WATAUAVAWH)xIHD)@At H  ~ H H fWpHt$PHHH HHD$HHLHHD$XHl$`oLHu H PsHSHD$ Hu H UH5LHu H 9HLHu H HHHD$8Hu H 9E3E3҃|vFIIMI+DD L@AH IA^ED IA^ML AA^ED IA^ML uD;s*MŋK M+A+ADHHA^EAuDsAfA(: THL$ FHDŽ$\ A=fWA$&HHMI+֋M+)$HT$@IAH+LHL$0HT$(H+HDE3BT(3fWAAAIIOI+DDHfDD H IYA\L YI\D YA\L YI\uE;},MAI M+A+BHHYA\uE3E3A|[AAIINI+DDLHDH IHAHDHAHDHAHDHAuE;}!MAK M+A+JHHHAuAyD^+A$;AHD$(IIL$LTAAI+DDH4H I I^+AYB0XD A ^ +AYJ(XL I^+AYB XD A ^ +AYJXL IuD;}GH$MDH+M+I ITE+HHI^+YBBXD Au4+fA()HT$(HL$@^H$HDHHL$0D\t5Ll$XH|$HHt$P($Hl$`HL$ D(D$p($Ht$HHHXuAENfH*HL$8YXCI;I3HL$ )IL$I[0Ik@IsHIA_A^A]A\_% % % % % % % %. @SH  HH HHHuC#H#H H 3H [HHXHhHxL` AUAVAWH 3ML8Q#D>eH%0HXH;t/ 3H4uAtcH  LHH MLHHI;rZH9}tH9EtHMHHEH H HL;uL;tLLIHpHq=SEH=K3eH%0HXH;t3H ut ;>HH uHH u HHH9=t!H tMĺIOH\$@Hl$HH|$PLd$XH A_A^A]HHXHpHxATH0ILXu9u 3ۉXtu7H HtЋ؉D$ LƋI0؉D$ LƋI؉D$ u5u1L3IL3ILMt L3IAӅtu7LƋI#ˋىL$ tH^HtLƋIЋ؉D$ 3ۉ\$ H\$@Ht$HH|$PH0A\H\$Ht$WH IHuoLNjHH\$0Ht$8H _H @SH HH HD$8Hu H~H |HD$8H xjHD$@HTHLD$@HT$8HHL$84HEHL$@"H+WHH [H(GHH(H\$WH HH=HHtHH;rH\$0H _H\$WH HH=HHtHH;rH\$0H _HMZf9t3HcH PAPADDINGXXPADDINGPADDINGXXPADDINGPADDINGXXPADDINGPADDINGXXPADDINGPADDINGXXPAD (PK C|7toeblitz-1.0/invToeplitz.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This is simply a wrapper function that calls decompToeplitz in a way that produces both ld, the log determinant of T, and Ti, the inverse. Note that the log determinant is computed essentially for free when the inverse is also obtained. We can specify a runMode here to run decompToeplitz in the desired mode. Here are the accepted values of runMode: % runMode = 0 % This calls the C/MEX implementation of Zohar's modification of the Trench algorithm. % runMode = 1 % This calls the backup Matlab implementation of Zohar's modification of the Trench algorithm. % runMode = -1 % This calls the generic MATLAB function logdet. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function[Ti, ld] = invToeplitz(T_row, runMode) if nargin < 2 % Handles the case when runMode is not specified. [ld, Ti] = decompToeplitz(T_row, 0); % This allows decompToeplitz to automatically choose the best method. else % Handles the case when runMode is specified. [ld, Ti] = decompToeplitz(T_row, 0, runMode); % This forces decompToeplitz to use the specified method. end end PK Ci}}toeblitz-1.0/license.txtGNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright © 2007 Free Software Foundation, Inc. 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Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This is simply a wrapper function that calls decompToeplitz in a way that computes only the log determinant, and does not waste computations and memory constructing and storing the inverse. We can specify a runMode here to run decompToeplitz in the desired mode. Here are the accepted values of runMode: % runMode = 0 % This calls the C/MEX implementation of Zohar's modification of the Trench algorithm. % runMode = 1 % This calls the backup Matlab implementation of Zohar's modification of the Trench algorithm. % runMode = -1 % This calls the generic MATLAB function logdet. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function[ld] = logdetToeplitz(T_row, runMode) if nargin < 2 % Handles the case when runMode is not specified. ld = decompToeplitz(T_row, 1); % This allows decompToeplitz to automatically choose the best method. else % Handles the case when runMode is specified. ld = decompToeplitz(T_row, 1, runMode); % This forces decompToeplitz to use the specified method. end endPK CTCk00toeblitz-1.0/README==================================================================== William Zhang, Washington University in St. Louis John P. Cunningham, Columbia University Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices Copyright (C) 2013 William B. Zhang and John P. Cunningham This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . ==================================================================== ==================================================================== Basic Usage Example ==================================================================== Here, we present a basic example of how to use our Toeblitz toolkit, with some random matrices and variables: T_row = genPDToeplitz(50); % This creates the variable T_row, the first row of a random 50 x 50 symmetric positive definite Toeplitz matrix T, which we will study. A = randn(50, 50); % This creates a random 50 x 50 matrix to use in computing the trace of a product of a Toeplitz matrix with an arbitrary matrix. b = randn(50, 1); % This creates a random 50 x 1 vector to use in the matrix Tx = b for testing solveToeplitz. tr = traceToeplitz(A, T_row); % This computes the trace of the product of A and T in a fast way. [invT, ldT] = invToeplitz(T_row); % This computes the inverse of T and the log determinant of T in a fast way. x = solveToeplitz(T_row, b); % This solves the system Tx = b for x. ==================================================================== ==================================================================== C/MEX: MEX functions are programs written in C, C++, or FORTRAN that are callable from Matlab. This allows bottlenecks in native Matlab code to be optimized through the use of a compiled language. In this package, we have pre-compiled Matlab MEX functions from C code for several popular operating systems. If we do not have a precompiled version for your operating system, the startup.m file will automatically attempt create one for your operating system. This requires a C compiler. Supported compilers can be found at . More information on C/MEX can be found at . Note: We have also pre-compiled MEX files for Octave, but have only done so (and tested our C functions) on an Ubuntu distribution of Linux. ==================================================================== ==================================================================== Startup.m This script will run by default if Matlab starts in the /toeplitz directory. If Matlab does not start in the /toeplitz directory, you should switch to it using "cd .../toeplitz", where .../toeplitz is the full directory path of the directory on your system. Then, you should run startup.m manually by simply typing "startup" into Matlab. It adds the locations of all the functions contained in our script to Matlab's search path, and attempts to create the relevant MEX files for your operating system, if it does not already exist. ==================================================================== ==================================================================== solveToeplitz.m solveToeplitz allows us to solve a Toeplitz system Tx = b for the unknown vector x. We call solveToeplitz with the following syntax: x = solveToeplitz(T_row, b [, runMode]) where b is the column vector b from our equation, T_row is the first row of the Toeplitz matrix T, x is the desired vector from our equation, and runMode is an optional argument forcing the code to use a particular method to obtain the solution x. Valid values of runMode are presented in the documentation in the main solveToeplitz function. ==================================================================== ==================================================================== logdetToeplitz.m logdetToeplitz allows us to compute the log determinant of a Toeplitz matrix T. logdetToeplitz is faster than obtaining the log determinant as a secondary output from invToeplitz, since it wastes no memory constructing the full inverse. It is linear in memory and quadratic in runtime. The calling syntax for logdetToeplitz is ldT = logdetToeplitz(T_row [, runMode]) where ldT is the log determinant of T, T_row is the first row of the Toeplitz matrix T, and runMode is an optional argument forcing the code to use a particular method to obtain the log determinant ldT. Valid values of runMode are presented in the documentation in the main logdetToeplitz function. ==================================================================== ==================================================================== traceToeplitz.m This function takes the trace of the product of a Toeplitz matrix T and an arbitrary matrix A in a smart way. The calling syntax is tr = traceToeplitz(A, T_row [, T_col, runMode]) where A is the arbitrary matrix, T_row is the first row of T, tr is the trace of the product A*T, T_col is an optional argument giving the first column of T (when T is asymmetric), and runMode is an optional argument forcing the code to use a particular method to obtain the trace. Valid values of runMode are presented in the documentation in the main traceToeplitz function. Note that in our tests, trace(A*T) is used as the MATLAB built-in function. sum(sum(transpose(A).*T)) is actually equivalent, and gives the answer faster than trace(A*T). However, our implementation exploits Toeplitz structure to perform the computation in the minimum (n^2 + n - 1) floating point operations, whereas sum(sum(transpose(A).*T)) requires (2n^2 - 1) operations. In our analysis we choose trace(A*T) as the comparison in order to illustrate the contrast between its O(n^3) complexity and the O(n^2) complexity of our traceToeplitz method. ==================================================================== ==================================================================== invToeplitz.m invToeplitz computes the inverse of a Toeplitz matrix T, and gives the log determinant of T for free in an optional second output argument. The calling syntax for invToeplitz is [invT [, ldT]] = invToeplitz(T_row [, runMode]) where invT is the inverse of T, ldT is an optional output argument giving the log determinant of T, T_row is the first row of the Toeplitz matrix T, and runMode is an optional argument forcing the code to use a particular method to obtain the inverse invT. Valid values of runMode are presented in the documentation in the main invToeplitz function. ==================================================================== ==================================================================== testToeplitz.m You can use this script to test the accuracy and speed of the supported functions in this package. The calling syntax is testToeplitz(method, sizes_vec, runs, save_flag)". where method is a string giving the name of the method to be tested (valid values of method are described in the documentation of the main function, testToeplitz), sizes_vec is a row vector of matrix sizes for which to test that method, runs is the number of iterations to average over for each data point, and save_flag is a Boolean indicating whether the results of the test should be saved, or simply displayed. To display the results without saving, save_flag can be set to 0 (false), or simply omitted. To save your results, save_flag must be set to 1. In this case, the figures will not be displayed, but will be saved as both .pdf files and .fig files to /toeplitz/results/XXX/, where XXX is a number assigned to that set of data, ranging from 001 to 999. ==================================================================== ==================================================================== recreateToeblitzPlots.m This script, when run with no input parameters, will generate and save to disk the three plots presented in the Toeblitz paper. It should be able to complete running overnight on a modern laptop. For a quicker verification, the -1 argument can be used, as follows: recreateToeblitzPlots(-1) Other options exist to only produce one of the three plots in the Toeblitz paper, and these are documented in the recreateToeblitzPlots.m file itself. ==================================================================== ==================================================================== convertToeplitz.m convertToeplitz is provided in the case that the user wishes to convert between our abbreviated representation of a Toeplitz matrix as its first row (and sometimes also its first column, in the asymmetric case) and the full explicit representation. Its calling syntax is T = convertToeplitz(T_row [, T_col]) [T_row [, T_col]] = convertToeplitz(T) where T is the explicit full representation of the Toeplitz matrix, and T_row and T_col are the first row and column, respectively. We see that T_col is an optional input used to specify asymmetric Toeplitz matrices, and that it is only given as an output when an asymmetric Toeplitz matrix is given as the input argument. ==================================================================== ==================================================================== The BibTeX citations for the primary papers used in this project are below: @book{golub1996matrix, title={Matrix computations}, author={Golub, Gene H and Van Loan, Charles F}, volume={3}, year={2012}, publisher={The Johns Hopkins University Press} } @book{rasmussen2006gaussian, title={Gaussian processes for machine learning}, author={Rasmussen, Carl E and Williams, Christopher KI}, year={2006}, publisher={MIT Press} } @article{levinson1947wiener, title={The {W}iener {R}{M}{S} (root mean square) error criterion in filter design and prediction}, author={Levinson, Norman}, journal = {Journal of Mathematics and Physics}, volume={25}, number={4}, pages={261–-278}, year={1947} publisher={MIT Press} } @article{durbin1960fitting, title={The fitting of time-series models}, author={Durbin, James}, journal={Revue de l'Institut International de Statistique}, pages={233--244}, year={1960}, publisher={JSTOR} } @article{holmes1989random, title={On random correlation matrices {I}{I}: the {T}oeplitz case}, author={Holmes, Robert B}, journal={Communications in Statistics-Simulation and Computation}, volume={18}, number={4}, pages={1511--1537}, year={1989}, publisher={Taylor \& Francis} } @article{zohar1969toeplitz, title={Toeplitz matrix inversion: the algorithm of {W}{F} {T}rench}, author={Zohar, Shalhav}, journal={Journal of the Association for Computing Machinery (JACM)}, volume={16}, number={4}, pages={592--601}, year={1969}, publisher={ACM} } @article{chan1996conjugate, title={Conjugate gradient methods for {T}oeplitz systems}, author={Chan, Raymond H and Ng, Michael K}, journal={SIAM review}, volume={38}, number={3}, pages={427--482}, year={1996}, publisher={SIAM} } @article{shewchuk1994introduction, title={An introduction to the conjugate gradient method without the agonizing pain}, author={Shewchuk, Jonathan R}, year={1994}, journal={CMU Technical Report} publisher={Carnegie Mellon University, Pittsburgh, PA} } @article{minka2003lightspeed, title={The lightspeed {MATLAB} toolbox}, author={Minka, Tom}, journal={Efficient operations for Matlab programming, Version}, volume={2}, year={2003} } ====================================================================PKϹ Ctoeblitz-1.0/results/PKu Etoeblitz-1.0/solve/PK C\ \ toeblitz-1.0/solveToeplitz.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This solves a system Tx = b for x, where x and b are column vectors and T is a toeplitz matrix. It uses the method specified by runMode, or chooses a method based on the size of the matrix and vector. % Accepted values of runMode are: % % runMode = 'LD' This directs the code to execute the Levinson-Durbin algorithm, which works in % runMode = 'PCG' This directs the code to execute the preconditioned conjugate gradient method with automatic choice of preconditioner. The code will always choose Chan. % runMode = 'Mldivide' This directs the code to use the built-in MATLAB mldivide method. % runMode = 'PCG_Chan' This explicitly directs the code to use the preconditioned conjugate gradient method with the Chan preconditioner. % runMode = 'PCG_Circulant_Embedding' This explicitly directs the code to use the preconditioned conjugate gradient method with a preconditioner produced by embedding our Toeplitz matrix in a circulant matrix and adding it to the other block. This is described more fully in the "Conjugate gradient methods for Toeplitz systems" 1996 article we reference in the paper accompanying this software package. % runMode = 'PCG_None' This explicitly directs the code to use the unconditioned conjugate gradient method. % % The calling syntax for solveToeplitz is % % solveToeplitz(T_row, b [, runMode]) % % where % % T_row is the first row of the Toeplitz matrix T % b is a the column vector % runMode (optional) is the method used to solve for x. % % The output of solveToeplitz is a single variable % % x, which is the solution to the matrix equation Tx = b. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [x] = solveToeplitz(T_row, b, runMode) size_T_row = size(T_row, 2); % Finds the size of the Toeplitz matrix. This is used for determining which method to use (different methods are fastest at different matrix sizes). if nargin < 3 if size_T_row < 500 runMode = 'mldivide'; % Uses Mldivide for smaller matrices. elseif size_T_row >= 500 runMode = 'pcg'; % Uses the Chan-preconditioned conjugate gradient method for larger matrices. end end x = solveToeplitzMode(T_row, b, runMode); end function [x] = solveToeplitzMode(T_row, b, runMode) %This function links the runMode handles to the actual function which implement the different methods. if strcmp(lower(runMode), 'ld') x = solveToeplitzLD(T_row, b); elseif strcmp(lower(runMode), 'mldivide') x = solveToeplitzMldivide(T_row, b); elseif strcmp(lower(runMode(1, 1:3)), 'pcg') if length(runMode) == 3 %This case allows our code to choose the preconditioner automatically. x = solveToeplitzPCG(T_row, b, 'auto'); elseif runMode(1, 4) == '_' %This case forces solveToeplitzPCG to run with a specified preconditioner. x = solveToeplitzPCG(T_row, b, runMode(5:end)); else error('Invalid runMode value given.') end else error('Invalid runMode value given.') end endPK Ci(toeblitz-1.0/solve/PCG_Chan.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This function builds the first column of the Chan circulant preconditioner as described in 2.1.2 (T. Chan's Preconditioner) of Conjugate gradient methods for Toeplitz systems (1996) by Raymond H. Chan and Michael K. Ng. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function[c] = PCG_Chan(T_row) n = size(T_row, 2); % Gets the size of the Toeplitz matrix we are dealing with. c(1, 1) = T_row(1, 1); % Obtains the first entry of our column vector directly from the Toeplitz matrix. % Non-vectorized version. %for i = 1:n-1 % Loops over all following entries. % c(i+1, 1) = ((n-i)*T_row(1, i+1) + i*T_row(1, n-i+1))/n; % Computes following entries according to equation 2.4 from Chan (1996). %end % Vectorized version. This should be somewhat faster, but is more costly in terms of memory requirements. n_vec = transpose(zeros(1, n-1)); % Initializes n_vec as a vector of all zeros. n_vec(1:n-1, 1) = n; % Sets the value of every entry of n_vec to n. i_vec = transpose([1:n-1]); % Creates a vector with the desired indices we want to loop over. c(2:n, 1) = ((n_vec-i_vec).*transpose(T_row(1, 2:n)) + i_vec.*transpose(T_row(1, n:-1:2)))/n; % % Computes following entries according to equation 2.4 from Chan (1996). endPK CL-ˆ,toeblitz-1.0/solve/PCG_Circulant_Embedding.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % Given a symmetric positive definite Toeplitz matrix, T, this function returns the first column of the circulant preconditioner formed by circulant embedding as in 2.1.3 (Preconditioners by embedding) of Conjugate gradient methods for Toeplitz systems (1996) by Raymond H. Chan and Michael K. Ng. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [r] = PCG_Circulant_Embedding(T_row) b = vertcat(T_row(1, 1), transpose(T_row(1, end:-1:2))); % Generate the first column of the matrix B, which is embedding in a circulant matrix along with our original matrix T. r = b + transpose(T_row(1, 1:end)); % Add the first column of T to the first column of B to generate the first column of our desired preconditioner. endPK Co8>>$toeblitz-1.0/solve/solveToeplitzLD.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This solves a system Tx = b for x, where x and b are column vectors and T is a positive definite Toeplitz matrix. It uses the Levinson-Durbin algorithm. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [x] = solveToeplitzLD(T_row, b) f_v = 1/T_row(1, 1); % This is the first forward vector. b_v = 1/T_row(1, 1); % This is the first backward vector. x = b(1, 1)/T_row(1, 1); % This is the first vector in the sequence x_n leading up to x_N = x, x_1. for i = 2:size(T_row, 2) %Obtaining the forward and backward vectors. f_error = T_row(1, i:-1:2)*f_v; %Computing the error terms that arise when we multiply T_n by the temporary vector. b_error = T_row(1, 2:i)*b_v; x_error = T_row(1, i:-1:2)*x; f_v = (1/(1-f_error*b_error))*(vertcat(f_v, 0) - f_error*vertcat(0, b_v)); %Computes new values for the forward and backwards vectors, as well as the next vector in the x_n sequence. b_v = f_v(end:-1:1); %b_v = (1/(1-f_error*b_error))*b_temp - (b_error/(1-f_error*b_error))*f_temp; %This is commented out, since we are working with symmetric matrices; so we can obtain the backward vector simply by reversing the forward vector. x = vertcat(x, 0) + (b(i, 1)-x_error)*b_v; %Ti*f_v % You can use this to check if the forward vectors are correct; this line of code should produce a vector with a one in the first entry and all zeroes following if they are. end endPK C*toeblitz-1.0/solve/solveToeplitzMldivide.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This solves a system Tx = b for x, where x and b are column vectors and T is a toeplitz matrix. It uses the built-in Matlab method. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [x] = solveToeplitzMldivide(T_row, b) x = mldivide(toeplitz(T_row), b); endPK Cs s %toeblitz-1.0/solve/solveToeplitzPCG.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This solves a system Tx = b for x, where x and b are column vectors and T is a positive definite toeplitz matrix. It uses the Conjugate Gradient method from page 529 of "Matrix Computations" by Golub. It also has a second (optional) output argument, k, which tells us the number of iterations it took to converge to our result. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [x, k] = solveToeplitzPCG(T_row, b, preconditioner) n = size(T_row, 2); % Get the size of the matrix we are dealing with. epsilon = 10^(-6); % Set a reasonable error bound for use in stopping condition. kmax = 400; % Set a maximum number of iterations. If it's taking more than 400, something is probably wrong. % Here we automatically set the preconditioner if not specified. if nargin < 3 preconditioner = 'auto'; end % Here we define our preconditioner, m. m is actually just the first column of our full circulant preconditioner M, but for speed we omit the rest of the matrix from explicit representation. if strcmp(lower(preconditioner), 'circulant_embedding') m = PCG_Circulant_Embedding(T_row); elseif or(strcmp(lower(preconditioner), 'chan'), strcmp(lower(preconditioner), 'auto')) m = PCG_Chan(T_row); elseif strcmp(lower(preconditioner), 'none') m = zeros(n, 1); m(1, 1) = 1; end if strcmp(lower(preconditioner), 'matlab') [x, foo1, foo2, k] = pcg(toeplitz(T_row), b); % Calls Matlab's default preconditioned conjugate gradient algorithm. We include this here primarily for testing purposes. Note foo used instead of ~ for octave compatibility else x = zeros(n, 1); % Here we choose an initial guess for x. The solution in this method is fairly insensitive to our initial guess, so we just choose the zero vector. k = 0; % Here we initialize k, the number of iterations, at zero. % Initialize the values of a bunch of other variables. r = b - multiplyToeplitz(T_row, x); d = multiplyinvCirculant(m, r); delta_new = transpose(r)*d; delta_old = delta_new; %The following loop runs the preconditioned conjugate gradient method. while and(sqrt(delta_old) > epsilon, k < kmax) q = multiplyToeplitz(T_row, d); alpha = delta_new/(transpose(d)*q); x = x + alpha*d; if mod(k, 50) == 0 r = b - multiplyToeplitz(T_row, x); else r = r - alpha*q; end s = multiplyinvCirculant(m, r); delta_old = delta_new; delta_new = transpose(r)*s; beta = delta_new/delta_old; d = s + beta*d; k = k + 1; end % Prints a warning message if the maximum number of iterations is reached. if k == kmax fprintf('WARNING: %d (maximum number) iterations taken, the residual is %g.', k, delta_old); end end end PK CWtoeblitz-1.0/startup.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This is a startup file which initializes Matlab's settings to allow for better use of the toeplitz package. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% path(sprintf('%s/util', pwd), path) % Adds the subdirectories of the root folder to the path, allowing us to call functions from them. path(sprintf('%s/test', pwd), path) path(sprintf('%s/decomp', pwd), path) path(sprintf('%s/solve', pwd), path) path(sprintf('%s/trace', pwd), path) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The following code checks for the relevant MEX files (such as .mexa64 % or .mexglx, depending on the machine architecture), and it creates the % mex file if it can not find the right one. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Create the mex file if necessary. startup_checkMEX('decomp', 'decompToeplitzFastZohar') % Toeplitz inversion startup_checkMEX('decomp', 'logdetToeplitzFastZohar') % Toeplitz log determinant startup_checkMEX('trace', 'traceToeplitzFast') % Toeplitz trace %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %This simply clears all variables and closes all windows opened by Matlab. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all % Clears all variables. close all % Closes all windows opened by Matlab.PKu Etoeblitz-1.0/test/PK CE|*--toeblitz-1.0/testToeplitz.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, John P Cunningham % This tests a toeplitz function with different sizes of matrices and times it to evaluate performance and accuracy. It takes two required input arguments, method and sizes_vec, and two optional arguments, runs and save_flag. The calling syntax is % testToeplitz(method, sizes_vec[, runs, save_flag]). % If three arguments are given, the code will assume that the third argument is runs. To specify save_flag, runs must also be specified. % % Valid inputs: % method = 'decompToeplitz', 'traceToeplitz', 'solveToeplitz', or 'solveToeplitzPCG' % sizes = a row vector of positive integer sizes over which to evaluate the performance of that function; for example, [5:10] will cause testToeplitz to produce results for matrices at sizes 5, 6, 7, 8, 9, and 10, and display relevant plots. % runs = a positive integer which indicates the number of runs over which to average results for each matrix size and algorithm. More runs will give smoother results and nicer graphs, but will be more computationally costly. This defaults to 3 if not specified. % save_flag = a boolean value, 0 or 1, which indicates whether plots should be displayed (0) or saved to the ./results directory. This defaults to 0 (only display, no saving) if not specified. % % Some example calls: % testToeplitz('decompToeplitz', [2:10:202]) % testToeplitz('traceToeplitz', [500:600]) % testToeplitz('solveToeplitz', [5 10 100 10^3 10^4], 10) % testToeplitz('solveToeplitzPCG', [400:100:1000], 1, 1) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This is the main function. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [] = testToeplitz(method, sizes_vec, runs, save_flag) % The following code sets default values for runs and save_flag if they are not specified. if nargin < 4 % Check to see if any arguments are missing. save_flag = 0; % Default value for save_flag is false. if nargin < 3 runs = 3; % Default number of runs to average over. end end % The following code finds a folder to save the result plots in, if save_flag is set to 1. folder = '0'; if save_flag for i = 1:999 % Searches from 001 to 999 for an unused folder name of the form 'test_XXX', where XXX is the three-digit number. if folder == '0' % Runs the following code if we have not yet found a suitable folder. test_folder = sprintf('./results/test_%03d/', i); % Finds the subdirectory in which the files will be placed. if not(exist(test_folder)) % Checks to see if there is already a folder with that name in the ./results/ directory. If not, it creates the folder and proceeds. folder = test_folder; % Changes the value of folder, stopping future checks for folder names. mkdir(folder); % Actually creates the directory. fprintf(['Printing to directory ' folder '.\n']) % Informs the user where the output will be saved. end end end end % The following code tests the accuracy and speed of our traceToeplitz method. Routines for testing other functions work similarly, and are less extensively commented. if strcmp(method, 'traceToeplitz') % This checks if 'traceToeplitz' is the method specified for testing. trace_strs = cell(4, 1); % This creates a cell array of the names of our different methods for use in labeling plots and legends. trace_strs(1, 1) = cellstr('trace(A*T)'); trace_strs(2, 1) = cellstr('traceToeplitz (Automated Choice)'); trace_strs(3, 1) = cellstr('traceToeplitz (C/MEX)'); trace_strs(4, 1) = cellstr('traceToeplitz (Matlab)'); vals_times = zeros(8, length(sizes_vec)); % This initializes the matrix in which all our test results will be (temporarily) stored. for i = 1:length(sizes_vec) % This loops over each the sizes of matrices we want to test. current_size_str = num2str(sizes_vec(i)); % Finds the current matrix size being worked on. fprintf(['Working on matrix size ' current_size_str '.\n']) % Informs the user what matrix size we're up to. new_vals_times = zeros(8, 1); % This initializes new_vals_times (and resets it for each size of matrix tested). for j = 1:runs % For each size matrix, this loops 'runs' times to obtain an average result. run_str = num2str(j); % Finds the current run being worked on. fprintf(['Now on run number ' run_str '.\n']) % Informs the user which run we are on. T_row = genPDToeplitz(sizes_vec(i)); % This produces the first row of a (more or less) random positive definite Toeplitz matrix. new_vals_times = new_vals_times + test_traceToeplitz(T_row); % This calculates the trace of the product with an arbitrary matrix in various ways, and adds the timings and accuracy metrics to a cumulative total in new_vals_times (see test_traceToeplitz documentation for more detail). end vals_times(1:8, i) = new_vals_times(1:8, 1)/runs; % This saves the results in new_vals_times to the appropriate column in vals_times, dividing by the number of runs to obtain an average. end % Show or save results % These are results for computational time. test_plot('all', sizes_vec, vals_times(2:2:8, 1:end), trace_strs, 'Computation time (s)', save_flag, folder, 'test_trace_time', 0); % These are results for accuracy. test_plot('linear', sizes_vec, vals_times(1:2:7, 1:end), trace_strs, 'Norm difference vs. trace(A*T)', save_flag, folder, 'test_trace_norm', 0); elseif strcmp(method, 'decompToeplitz') decomp_strs = cell(4, 1); decomp_strs(1, 1) = cellstr('decompToeplitz (Automated Choice)'); decomp_strs(2, 1) = cellstr('inv(T)'); decomp_strs(3, 1) = cellstr('C/MEX Zohar'); decomp_strs(4, 1) = cellstr('Matlab Zohar'); vals_times = zeros(12, length(sizes_vec)); for i = 1:length(sizes_vec) current_size_str = num2str(sizes_vec(i)); % Finds the current matrix size being worked on. fprintf(['Working on matrix size ' current_size_str '.\n']) % Informs the user what matrix size we're up to. new_vals_times = zeros(12, 1); for j = 1:runs run_str = num2str(j); % Finds the current run being worked on. fprintf(['Now on run number ' run_str '.\n']) % Informs the user which run we are on. T_row = genPDToeplitz(sizes_vec(i)); % Make random positive definite symmetric toeplitz matrix. % Mow calculate the inverse in various ways, and save the timings and accuracy metrics (see test_decompToeplitz documentation for more detail). new_vals_times = new_vals_times + test_decompToeplitz(T_row, i); end vals_times(1:12, i) = new_vals_times(1:12, 1)/runs; % Update our array of results with the average of all trials, after each size matrix in sizes_vec is tested. end % Show or save results % These are results for computational time. test_plot('all', sizes_vec, vals_times(3:3:12, 1:end), decomp_strs, 'Computation time (s)', save_flag, folder, 'test_decomp_time', 0); % These are results for accuracy. test_plot('linear', sizes_vec, vals_times(1:3:10, 1:end), decomp_strs, 'Frobenius norm difference vs. inv(T)', save_flag, folder, 'test_decomp_invnorm', 0); test_plot('linear', sizes_vec, vals_times(2:3:11, 1:end), decomp_strs, 'Norm difference vs. logdet(T)', save_flag, folder, 'test_decomp_logdetnorm', 0); elseif strcmp(method, 'solveToeplitz') solve_strs = cell(4, 1); solve_strs(1, 1) = cellstr('Matlab mldivide'); solve_strs(2, 1) = cellstr('Levinson-Durbin'); solve_strs(3, 1) = cellstr('Unconditioned Conjugate Gradient'); solve_strs(4, 1) = cellstr('Preconditioned Conjugate Gradient'); vals_times = zeros(8, length(sizes_vec)); for i = 1:length(sizes_vec) current_size_str = num2str(sizes_vec(i)); % Finds the current matrix size being worked on. fprintf(['Working on matrix size ' current_size_str '.\n']) % Informs the user what matrix size we're up to. new_vals_times = zeros(8, 1); for j = 1:runs run_str = num2str(j); % Finds the current run being worked on. fprintf(['Now on run number ' run_str '.\n']) % Informs the user which run we are on. % Make a random positive definite symmetric toeplitz matrix. T_row = genPDToeplitz(sizes_vec(i)); % Generates an arbitrary vector b to use in the equation Tx = b when solving for x. b = [1:sizes_vec(i)]; b = b(:); % Now solve the system Tx = b in various ways, and return the timings and accuracy metrics. new_vals_times = new_vals_times + test_solveToeplitz(T_row, b); %Tests the accuracy and speed of our algorithms for this size matrix. end vals_times(1:8, i) = new_vals_times(1:8, 1)/runs; % Adds the results from new_vals_times to val_times. end % Show or save results %These are results for computational time. test_plot('all', sizes_vec, vals_times(2:2:8, 1:end), solve_strs, 'Computation time (s)', save_flag, folder, 'test_solve_time', 0); %These are results for accuracy. test_plot('linear', sizes_vec, vals_times(1:2:7, 1:end), solve_strs, 'Norm difference vs. Built-in mldivide', save_flag, folder, 'test_solve_norm', 0); elseif strcmp(method, 'solveToeplitzPCG') PCG_strs = cell(4, 1); PCG_strs(1, 1) = cellstr('Unconditioned Conjugate Gradient'); PCG_strs(2, 1) = cellstr('MATLAB pcg'); PCG_strs(3, 1) = cellstr('Chan Preconditioner'); PCG_strs(4, 1) = cellstr('Circulant Embedding Preconditioner'); vals_times = zeros(12, length(sizes_vec)); for i = 1:length(sizes_vec) current_size_str = num2str(sizes_vec(i)); % Finds the current matrix size being worked on. fprintf(['Working on matrix size ' current_size_str '.\n']) % Informs the user what matrix size we're up to. new_vals_times = zeros(12, 1); for j = 1:runs run_str = num2str(j); % Finds the current run being worked on. fprintf(['Now on run number ' run_str '.\n']) % Informs the user which run we are on. % Make random positive definite symmetric toeplitz matrix. T_row = genPDToeplitz(sizes_vec(i)); % Generates an arbitrary vector b to use in the equation Tx = b when solving for x. b = [1:sizes_vec(i)]; b = b(:); % Now solve the system Tx = b in various ways, and return the timings and accuracy metrics. new_vals_times = new_vals_times + test_solveToeplitzPCG(T_row, b); %Tests the accuracy and speed of our algorithms for this size matrix. end vals_times(1:12, i) = new_vals_times(1:12, 1)/runs; % Adds the results from new_vals_times to val_times. end % Show or save results % These are results for computational time. test_plot('all', sizes_vec, vals_times(2:3:11, 1:end), PCG_strs, 'Computation time (s)', save_flag, folder, 'test_PCG_time', 0); % These are results for accuracy. test_plot('linear', sizes_vec, vals_times(1:3:10, 1:end), PCG_strs, 'Norm difference vs. Built-in mldivide', save_flag, folder, 'test_PCG_norm', 0); %These are results for the number of iterations. test_plot('linear', sizes_vec, vals_times(3:3:12, 1:end), PCG_strs, 'Number of Iterations', save_flag, folder, 'test_PCG_iterations', 0); else error('No such method found.') end end PK CԂx x 'toeblitz-1.0/test/test_decompToeplitz.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This function takes a Toeplitz matrix T and its dimension i as input, and returns a structured array with rows containing accuracy results and timings for various methods of calculating the inverse, and columns corresponding to i, the sizes of the different matrices tested. For each method, the upper row contains the accuracy results for the inverse, while the middle row contains the accuracy results for the log determinant, and the lower row contains the timings for the inverse. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function[new_vals_times] = test_decompToeplitz(T_row, i) new_vals_times = zeros(12, 1); % Initialize the vector of results with zeros. % Call decompToeplitz(T) and let the code figure out the best method automatically. t0 = clock; [TiInvT, ldT] = decompToeplitz(T_row, 0); new_vals_times(3, 1) = etime(clock,t0); % Saves the time taken to compute the inverse and log determinant in the third row containing information for this method. % Call to decompToeplitz(T,-1) and run the built-in Matlab functions, simple inv() and logdet(). t0 = clock; [TiInv, ld] = decompToeplitz(T_row, 0, -1); tInv(i) = etime(clock,t0); new_vals_times(6, 1) = etime(clock,t0); % Saves the time taken to compute the inverse and log determinant in the third row containing information for this method. % Call to decompToeplitz(T,0) and run the MEX implementation of Zohar. t0 = clock; [TiInvT0, ldT0] = decompToeplitz(T_row, 0, 0); new_vals_times(9, 1) = etime(clock,t0); % Saves the time taken to compute the inverse and log determinant in the third row containing information for this method. % Call to decompToeplitz(T,1) and run the native Matlab implementation of Zohar. t0 = clock; [TiInvT1, ldT1] = decompToeplitz(T_row, 0, 1); new_vals_times(12, 1) = etime(clock,t0); % Saves the time taken to compute the inverse and log determinant in the third row containing information for this method. new_vals_times(1, 1) = norm(TiInv(:) - TiInvT(:)); % Frobenius norm (The (:) linearizes the matrix into a vector and allows computation of the norm.) new_vals_times(4, 1) = norm(TiInv(:) - TiInv(:)); % Frobenius norm (The (:) linearizes the matrix into a vector and allows computation of the norm.) new_vals_times(7, 1) = norm(TiInv(:) - TiInvT0(:)); % Frobenius norm (The (:) linearizes the matrix into a vector and allows computation of the norm.) new_vals_times(10, 1) = norm(TiInv(:) - TiInvT1(:)); % Frobenius norm (The (:) linearizes the matrix into a vector and allows computation of the norm.) new_vals_times(2, 1) = norm(ld - ldT); % Since the ld is a scalar, the norm is much simpler to obtain. new_vals_times(5, 1) = norm(ld - ld); % Since the ld is a scalar, the norm is much simpler to obtain. new_vals_times(8, 1) = norm(ld - ldT0); % Since the ld is a scalar, the norm is much simpler to obtain. new_vals_times(11, 1) = norm(ld - ldT1); % Since the ld is a scalar, the norm is much simpler to obtain. endPK L C_  toeblitz-1.0/test/test_plot.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This function plots results, given in data_mat, on a log axis against the vector/matrix sizes, given in sizes_vec, on a linear axis. data_labels contains strings which are used to label to axes. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This main function serves primarily to parse the plot_mode input and use it to call test_plotmode appropriately. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [] = test_plot(full_plot_mode, sizes_vec, data_mat, data_labels, yaxis_label, save_flag, folder, file_name, fig_save) if strcmp(lower(full_plot_mode), 'all') test_plotmode('linear', sizes_vec, data_mat, data_labels, yaxis_label, save_flag, folder, file_name, fig_save); test_plotmode('semilogy', sizes_vec, data_mat, data_labels, yaxis_label, save_flag, folder, file_name, fig_save); test_plotmode('loglog', sizes_vec, data_mat, data_labels, yaxis_label, save_flag, folder, file_name, fig_save); else test_plotmode(lower(full_plot_mode), sizes_vec, data_mat, data_labels, yaxis_label, save_flag, folder, file_name, fig_save); end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % test_plotmode actually produces a plot from the input passed to it by test_plot. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [a_graph] = test_plotmode(plot_mode, sizes_vec, data_mat, data_labels, yaxis_label, save_flag, folder, file_name, fig_save) data_series_num = size(data_mat, 1); % Figures out how many data series are being plotted; in this package, it will always be 3 or 4. a_graph = figure; % Creates a figure. % If there are three data series, produce the appropriate figure based on plot_mode. if data_series_num == 3 if strcmp(lower(plot_mode), 'linear') plot(sizes_vec, data_mat(1, 1:end), 'r', sizes_vec, data_mat(2, 1:end), 'b', sizes_vec, data_mat(3, 1:end), 'm', 'LineWidth',3); elseif strcmp(lower(plot_mode), 'semilogy') semilogy(sizes_vec, data_mat(1, 1:end), 'r', sizes_vec, data_mat(2, 1:end), 'b', sizes_vec, data_mat(3, 1:end), 'm', 'LineWidth',3); elseif strcmp(lower(plot_mode), 'loglog') loglog(sizes_vec, data_mat(1, 1:end), 'r', sizes_vec, data_mat(2, 1:end), 'b', sizes_vec, data_mat(3, 1:end), 'm', 'LineWidth',3); end % If there are four data series, produce the appropriate figure based on plot_mode. elseif data_series_num == 4 if strcmp(lower(plot_mode), 'linear') plot(sizes_vec, data_mat(1, 1:end), 'r', sizes_vec, data_mat(2, 1:end), 'b', sizes_vec, data_mat(3, 1:end), 'm', sizes_vec, data_mat(4, 1:end), 'g', 'LineWidth',3); elseif strcmp(lower(plot_mode), 'semilogy') semilogy(sizes_vec, data_mat(1, 1:end), 'r', sizes_vec, data_mat(2, 1:end), 'b', sizes_vec, data_mat(3, 1:end), 'm', sizes_vec, data_mat(4, 1:end), 'g', 'LineWidth',3); elseif strcmp(lower(plot_mode), 'loglog') loglog(sizes_vec, data_mat(1, 1:end), 'r', sizes_vec, data_mat(2, 1:end), 'b', sizes_vec, data_mat(3, 1:end), 'm', sizes_vec, data_mat(4, 1:end), 'g', 'LineWidth',3); end end % Label the axes appropriately and set the font to a fixed size. xlabel('Matrix Size','FontSize',18); ylabel(yaxis_label,'FontSize',18); set(gca,'fontsize',18); % Label the legend appropriately based on the number of data series. if data_series_num == 3 legend(char(data_labels(1, 1)), char(data_labels(2, 1)), char(data_labels(3, 1))); elseif data_series_num == 4 legend(char(data_labels(1, 1)), char(data_labels(2, 1)), char(data_labels(3, 1)), char(data_labels(4, 1))) end % Either saves or displays the plot, based on the value of save_flag. if save_flag print(a_graph, strcat(folder, file_name), '-dpdf'); if fig_save print(a_graph, strcat(folder, file_name), '-dfig'); end close all else a_graph; end end PK C_ug&toeblitz-1.0/test/test_solveToeplitz.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This function takes an n x n Toeplitz matrix T and a nx1 column vector b as input, and returns a structured array with rows containing accuracy results and timings for various methods of calculating solving the system Tx = b for x, and columns corresponding to i, the sizes of the different matrices tested. For each method, the upper row contains the accuracy results for the solution x, while the lower row contains the timings for the solution. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [new_vals_times] = test_solveToeplitz(T_row, b) new_vals_times = zeros(6, 1); % Initialize the vector of results with zeros. % Call solveToeplitz and specify the default Matlab mldivide algorithm. t = cputime; x = solveToeplitz(T_row, b, 'Mldivide'); new_vals_times(2, 1) = cputime-t; % Times the default algorithm. new_vals_times(1, 1) = norm(x(:) - x(:)); % Compares accuracy to built-in Matlab algorithm. % Call solveToeplitz and specify the Levinson-Durbin algorithm. t = cputime; xLD = solveToeplitz(T_row, b, 'LD'); new_vals_times(4, 1) = cputime-t; % Times my algorithm. new_vals_times(3, 1) = norm(x(:) - xLD(:)); % Compares accuracy to built-in Matlab algorithm. % Call solveToeplitz and specify the Conjugate Gradient method. t = cputime; xCG = solveToeplitz(T_row, b, 'PCG_None'); new_vals_times(6, 1) = cputime-t; % Times my algorithm. new_vals_times(5, 1) = norm(x(:) - xCG(:)); % Compares accuracy to built-in Matlab algorithm. % Call solveToeplitz and specify the Preconditioned Conjugate Gradient method with automatic choice of preconditioner. t = cputime; xPCG = solveToeplitz(T_row, b, 'PCG_Auto'); new_vals_times(8, 1) = cputime-t; % Times my algorithm. new_vals_times(7, 1) = norm(x(:) - xPCG(:)); % Compares accuracy to built-in Matlab algorithm. endPK Cw )toeblitz-1.0/test/test_solveToeplitzPCG.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This function takes an n x n Toeplitz matrix T and a nx1 column vector b as input, and returns a structured array with rows containing accuracy results and timings for various preconditioners used for solving the system Tx = b for x by the conjugate gradient method, and columns corresponding to i, the sizes of the different matrices tested. For each method, the upper row contains the accuracy results for the solution x, while the middle row contains the timings for the solution, and the lower row contains the number of iterations required for the solution to converge. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [new_vals_times] = test_solveToeplitzPCG(T_row, b) new_vals_times = zeros(12, 1); % Intialize new_vals_times as a vector of zeros. x = solveToeplitzMldivide(T_row, b); % Get a standard against which to measure the accuracy of our preconditioned conjugate gradient methods. % Call solveToeplitzPCG and specify the unconditioned conjugate gradient method. t = cputime; [xPCG, k] = solveToeplitzPCG(T_row, b, 'None'); new_vals_times(2, 1) = cputime-t; % Times the default algorithm. new_vals_times(1, 1) = norm(x(:) - xPCG(:)); % Compares accuracy to built-in Matlab algorithm. new_vals_times(3, 1) = k; % Stores the number of iterations required to converge. % Call solveToeplitzPCG and specify built-in MATLAB preconditioned conjugate gradient method. t = cputime; [xPCG, k] = solveToeplitzPCG(T_row, b, 'Matlab'); new_vals_times(5, 1) = cputime-t; % Times the default algorithm. new_vals_times(4, 1) = norm(x(:) - xPCG(:)); % Compares accuracy to built-in Matlab algorithm. new_vals_times(6, 1) = k; % Stores the number of iterations required to converge. % Call solveToeplitzPCG and specify the Chan preconditioned conjugate gradient method. t = cputime; [xPCG, k] = solveToeplitzPCG(T_row, b, 'Chan'); new_vals_times(8, 1) = cputime-t; % Times the default algorithm. new_vals_times(7, 1) = norm(x(:) - xPCG(:)); % Compares accuracy to built-in Matlab algorithm. new_vals_times(9, 1) = k; % Stores the number of iterations required to converge. % Call solveToeplitzPCG and specify the Circulant Embedding preconditioned conjugate gradient method. t = cputime; [xPCG, k] = solveToeplitzPCG(T_row, b, 'Circulant_Embedding'); new_vals_times(11, 1) = cputime-t; % Times the default algorithm. new_vals_times(10, 1) = norm(x(:) - xPCG(:)); % Compares accuracy to built-in Matlab algorithm. new_vals_times(12, 1) = k; % Stores the number of iterations required to converge. endPK Ctrhuu&toeblitz-1.0/test/test_traceToeplitz.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This compares the performance of traceToeplitz.m to the built-in trace function for a matrix of size n. For each method, the first row (among the rows its data occupies) contains the norm of the difference between a method and explicitly computing the product and then taking the trace, and the second contains the computational time used. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [new_vals_times] = test_traceToeplitz(T_row) n = size(T_row, 2); new_vals_times = zeros(8, 1); % Initializes new_vals_times. It is faster to modify a matrix than to concatenate two. A = randn(n, n); % This line generates an arbitrary random matrix, A, to use in computing the trace of the product of a Toeplitz matrix with an arbitrary matrix. t = cputime; tr = traceToeplitz(A, T_row, T_row, -1); new_vals_times(2, 1) = cputime-t; % Times the built-in Matlab algorithm. new_vals_times(1, 1) = norm(tr - tr); % Compares accuracy to built-in Matlab algorithm. t = cputime; trT = traceToeplitz(A, T_row, T_row, 0); new_vals_times(4, 1) = cputime-t; % Times my algorithm with automated choice of implementation. new_vals_times(3, 1) = norm(tr - trT); % Compares accuracy to built-in Matlab algorithm. t = cputime; trC = traceToeplitz(A, T_row, T_row, 1); new_vals_times(6, 1) = cputime-t; % Times my algorithm with the fast C implementation. new_vals_times(5, 1) = norm(tr - trC); % Compares accuracy to built-in Matlab algorithm. t = cputime; trM = traceToeplitz(A, T_row, T_row, 2); new_vals_times(8, 1) = cputime-t; % Times my algorithm with the backup Matlab implementation. new_vals_times(7, 1) = norm(tr - trM); % Compares accuracy to built-in Matlab algorithm. endPKu Etoeblitz-1.0/trace/PK Cѿ**toeblitz-1.0/traceToeplitz.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This generates the trace of the product of a Toeplitz matrix T, and an % arbitrary square matrix of compatible size, A. A wrapper function tries to use the C implementation, % but falls back on a Matlab implementation if that fails for any reason. % % The calling syntax is % % traceToeplitz(A, T_row, T_col, runMode) % % where % A is the arbitrary matrix, % T_row is the first row of the Toeplitz matrix, % T_col is the (optional) first column of the Toeplitz matrix, written as a row vector, and % runMode is an optional argument, giving the method by which to calculate the trace, with the following accepted values: % % runMode = -1 This calls the default MATLAB matrix multiplication and trace functions. It's very slow and should only be used for testing. % runMode = 0 This is the default method, which first tries to use the C/MEX implementation of our algorithm, but uses the backup MATLAB code if it fails. % runMode = 1 This is for testing only. It forces the code into using the C/MEX implementation of our algorithm with no backup. % runMode = 2 This forces the code into using the MATLAB implementation of our algorithm with no backup. % % traceToeplitz has a single output argument, % % Tr, which gives the desired trace. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This function mainly serves to parse the input. A, the arbitrary matrix, and T_row, the first row of our Toeplitz matrix, are both required. However, T_col, the first column of our Toeplitz matrix, is not required. When it is absent, we simply assume that the matrix is symmetric. In addition, runMode, which specifies which method to use to compute the trace, is optional. It defaults to case 0, which attempts to use fast C code, and resorts to a Matlab backup if the C code fails. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [ Tr ] = traceToeplitz(A, T_row, T_col_or_perhaps_runMode, runMode) if nargin < 2 % In this case, we have too few input arguments, and at least one required argument is missing. error('Not enough input arguments') elseif nargin == 2 % In this case, both T_col and runMode are empty. So we assume the Toeplitz matrix is symmetric and use the default case 0 runMode. T_col = T_row; runMode = 0; elseif nargin == 3 % In this case, we need to determine whether the optional argument given is a runMode or a T_col. if size(T_row) == [1 1] % If we are looking at trivial 1x1 matrices, then it is difficult to distinguish T_col from runMode, and therefore we do not allow omitted arguments in this case. error('Not enough input arguments. For trivial 1x1 matrices, T_row, T_col, and runMode must all be explicitly specified to prevent confusion.') elseif size(T_row) == size(T_col_or_perhaps_runMode) % If T_row is the same size as the given optional argument, we can safely assume that it is the first column of our Toeplitz matrix. T_col = T_col_or_perhaps_runMode; runMode = 0; else % Otherwise, the third argument given must be the runMode. T_col = T_row; runMode = T_col_or_perhaps_runMode; end elseif nargin == 4; T_col = T_col_or_perhaps_runMode; end Tr = traceToeplitzMode(A, T_row, T_col, runMode); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This function calls the actual code which will do computations based on the runMode passed to it by traceToeplitz. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [ Tr ] = traceToeplitzMode(A, T_row, T_col, runMode) switch(runMode) case -1 % Trivial case: this uses built-in Matlab functions to compute the trace of the product. This is intended primarily for testing. T = toeplitz(T_col, T_row); Tr = trace(A*T); case 0 % Attempts to use the faster C code, but defaults back to the backup Matlab code if anything goes wrong. This is the default case. try % Here it attempts to call the MEX function compiled from the C source code. Tr = traceToeplitzFast(A, T_row, T_col); catch % Here it falls back on the native Matlab code when the C code fails. Tr = traceToeplitzBackup(A, T_row, T_col); end case 1 % Forces the use of the C code. If it fails, then it really fails. This mode is intended mostly for testing. Tr = traceToeplitzFast(A, T_row, T_col); case 2 % Forces the use of the Matlab code. This is more reliable across systems since Matlab is an interpreted language, but is slower. This is intended mostly for testing. Tr = traceToeplitzBackup(A, T_row, T_col); end endPK C`y\jj(toeblitz-1.0/trace/traceToeplitzBackup.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % % This generates the trace of the product of a Toeplitz matrix and any % arbitrary square matrix of compatible size. This is a backup function % in case the C/MEX implementation fails for any reason. This function % computes the trace from the contributions from different parts of the % original matrices. In this case both the first row and column vectors % have been specified. % % The calling syntax is % % traceToeplitzFast(A, T_row, T_col) % % where % A is the arbitrary matrix, % T_row is the first row of the Toeplitz matrix, and % T_col is the first column of the Toeplitz matrix, written as a row vector. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [ Tr ] = traceToeplitzBackup(A, T_row, T_col) Tr = 0; % Initializes the trace at zero. for i = 2:size(T_col, 2), Tr = Tr + T_col(1, i)*sum(diag(A, i-1)); % Adds the contributions from each diagonal of the bottom half of the toeplitz matrix. Tr = Tr + T_row(1, i)*sum(diag(A, -i+1)); % Adds the contributions from each diagonal of the upper triangle of the toeplitz matrix. end Tr = Tr + T_row(1, 1)*sum(diag(A)); % Adds the contributions from the main diagonal of the toeplitz matrix. endPK Cdv &toeblitz-1.0/trace/traceToeplitzFast.c/*================================================================= * Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices * Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) *=================================================================*/ /* William Zhang, 2013 * This simple C function simply computes the trace of the product of a Toeplitz matrix T and an arbitrary matrix A. * Mathematically, it is identical to traceToeplitzBackup.m. The calling syntax is * * traceToeplitzFast(A, T_row, T_col) * * where * A is the arbitrary matrix, * T_row is the first row of the Toeplitz matrix, and * T_col is the first column of the Toeplitz matrix, written as a row vector. */ #include #include /* Input Arguments */ #define A_IN prhs[0] #define T_row_IN prhs[1] #define T_col_IN prhs[2] /* Output Arguments */ #define Tr_OUT plhs[0] void mexFunction( int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[] ) { double *A, Tr, *T_row, *T_col, temp_diag_sum, *Tr_Middle; int n, i, j; /*Get the size of the matrices we're dealing with.*/ n = mxGetN(A_IN); /* Create a scalar for the return argument */ Tr_OUT = mxCreateDoubleScalar(0); /*Assign pointers to variables we will use to pass along inputs and outputs.*/ A = mxGetPr(A_IN); T_row = mxGetPr(T_row_IN); T_col = mxGetPr(T_col_IN); Tr_Middle = mxGetPr(Tr_OUT); /*Add the contribution from the top half of the Toeplitz matrix (and therefore the bottom half of the arbitrary matrix A), not counting the principal diagonal.*/ for (i = 1; i < n; i++) { temp_diag_sum = 0; for (j = 0; j < i; j++) { temp_diag_sum += A[(n-1)*n - (i-1)*n + j*(n+1)]; } Tr += temp_diag_sum*T_row[n-i]; } /*Add the contribution from the bottom half of the Toeplitz matrix (and therefore the top half of the arbitrary matrix A), not counting the principal diagonal.*/ for (i = 1; i < n; i++) { temp_diag_sum = 0; for (j = 0; j < i; j++) { temp_diag_sum += A[n-1 - (i-1) + j*(n+1)]; } Tr += temp_diag_sum*T_col[n-i]; } /*Add the contribution from the principal diagonal of the Toeplitz matrix defined by T_row and coL_vec, and the principal diagonal of arbitrary matrix A.*/ for (i = 0; i < 1; i++) { temp_diag_sum = 0; for (j = 0; j < n; j++) { temp_diag_sum += A[j*(n+1)]; } Tr += T_row[0]*temp_diag_sum; *Tr_Middle = Tr; } return; } PK CȒ""(toeblitz-1.0/trace/traceToeplitzFast.mexELF44 (# HH$$QtdRtdttGNURXRt0ο 'ұSa     `  |CEqX4 + "gKRm Z a ?z l x __gmon_start___init_fini__cxa_finalize_Jv_RegisterClassesmexFunctionmxGetNmxCreateDoubleScalarmxGetPrliboctinterp.soliboctave.solibcruft.soliblapack.so.3gflibblas.so.3gflibfftw3.so.3libfftw3f.so.3libreadline.so.6libncurses.so.5libdl.so.2libhdf5.so.6libz.so.1libgfortran.so.3libstdc++.so.6libm.so.6libgcc_s.so.1libc.so.6_edata__bss_start_endGLIBC_2.1.3Psi r      US[|tX[ hhhhh UVS$u]t $()9s ((9rƃ$[^]US.ktt $Ѓ[]Ë$ÐUWVS\u!]} $$E]$EԋF$E̋F$Eċ$}EEEE}̃EЋEMЃU׋UEډU܋Uԍ‰E䍶U19|E܃E9uuًUԾ}U}؉UU19|m9uuًU1Mut&9uŰE \[^_]ËE}EِUVSmêt&Ћu[^]US[ppY[o)8BP l xo`l ~ (D$ oooo GCC: (Ubuntu 4.4.3-4ubuntu5.1) 4.4.3ztmexFunctionp!: int5upk:P(P,(\y AmuT Tr6\ muL muD 6[ amuH n( i( j( V = b h = 6% $ > $ > : ; $ > .? : ; ' @: ; I : ; I 4: ; I 4: ; I 4: ; I  I &I /home/cunni/toeplitz/wemmick/toeplitz/trace/usr/include/octave-3.2.3/octavetraceToeplitzFast.cmex.h7AU?עp@mmf>>,jy.y.mJb.>,kx.x.b-<>,kx | zCB Ilong long intunsigned charprhs/home/cunni/toeplitz/wemmick/toeplitz/trace/traceToeplitzFast.cTr_Middlelong long unsigned intmxArrayGNU C 4.4.3nlhsT_colT_rowplhsshort unsigned intlong doublenrhsshort intmexFunctiontemp_diag_sumttzuW2u 24W4gu gzW V2u24V4gugzVAdV2V4AVP2P4YP.symtab.strtab.shstrtab.note.gnu.build-id.gnu.hash.dynsym.dynstr.gnu.version.gnu.version_r.rel.dyn.rel.plt.init.text.fini.eh_frame.ctors.dtors.jcr.dynamic.got.got.plt.data.bss.comment.debug_aranges.debug_pubnames.debug_info.debug_abbrev.debug_line.debug_frame.debug_str.debug_loc$2H.o``<8 @ll~HoUo d $$ m DD( vll0q`|xxxH    0%= ]"t 4"0- 8"3 p `l$D l   x    (5 K Z h t@   x  ' , ; B V"rz ~l crtstuff.c__CTOR_LIST____DTOR_LIST____JCR_LIST____do_global_dtors_auxcompleted.7021dtor_idx.7023frame_dummy__CTOR_END____FRAME_END____JCR_END____do_global_ctors_auxtraceToeplitzFast.c__i686.get_pc_thunk.bx_GLOBAL_OFFSET_TABLE__DYNAMIC__DTOR_END____dso_handle_fini__bss_start_end__gmon_start___edata_Jv_RegisterClasses__cxa_finalize@@GLIBC_2.1.3mexFunctionmxGetPr_initmxGetNmxCreateDoubleScalarPK C@ ==+toeblitz-1.0/trace/traceToeplitzFast.mexa64ELF>@@@ @8@ll pp p    $$PtdQtdGNUh.%ca:UCOo@  5 [?  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Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This function automatically converts a Toeplitz matrix into a row vector containing all necessary information, or expands that row vector into a full Toeplitz matrix, depending on the input. For non-symmetric matrices, this function will output both the first row and first column. % % The calling syntax is % % T_row = convertToeplitz(T), where T is a symmetric Toeplitz matrix and T_row is its first row. % [T_row, T_col] = convertToeplitz(T), where T is an asymmetric Toeplitz matrix, T_row is its first row, and T_col is its first column. % T = convertToeplitz(T_row), where T_row is the first row of a symmetric Toeplitz matrix, and T is the full matrix. % T = convertToeplitz(T_row, T_col), where T_row is the first row of an asymmetric Toeplitz matrix T, T_col is the first column of that same matrix, and T is the full matrix. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [alt_form, alt_form2] = convertToeplitz(one_form, one_form2) if nargin == 1 % Fill in a value for one_form2 when it is not specified. one_form2 = 0; % Create a placeholder value for one_form2 when it is not specified in the input. This value is not used. end if size(one_form, 1) == size(one_form, 2) % This is the case in which the first input was a Toeplitz matrix. We only check that it is a square matrix, for the sake of efficiency. We leave it to the caller to make sure that the input is Toeplitz. alt_form = one_form(1, 1:end); if not(isequal(transpose(one_form(1:end, 1)), one_form(1, 1:end))) % This is the case in which the input was a non-symmetric Toeplitz matrix. alt_form2 = one_form(1:end, 1); % In this case we also output the first column of the original matrix as a second row in our output. end elseif size(one_form, 1) == 1 % This is the case in which the first input was a row vector. if nargin == 2 % If there were two input arguments, the Toeplitz matrix is not necessary symmetric. alt_form = toeplitz(one_form2, one_form); % We use the convention opposite that of the built-in toeplitz() method. So we reverse the order of our row and column vector to generate the correct Toeplitz matrix. elseif nargin == 1 % If there is only one input, generate a symmetric Toeplitz matrix. alt_form = toeplitz(one_form); % Generate a symmetric Toeplitz matrix. else error('Too many input arguments!') end else error('Invalid input!') end endPK C|1u !toeblitz-1.0/util/genPDToeplitz.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This function generates the first row of a random symmetric positive definite Toeplitz matrix of size n. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This is the main function. It generates a random positive definite Toeplitz matrix of size n from the algorithm given on page 4 of "On random correlation matrices: II. The Toeplitz case" (1989) by R. B. Holmes at the Lincoln Laboratories at the Massachusetts Institute of Technology. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [T_row] = genPDToeplitz(n) % Generate a random vector from the uniform distribution on the unit n-sphere. rand_vec = randn(n, 1); % Generate a random n-dimensional vector. This essentially generates the direction of our vector. rand_vec = rand_vec/norm(rand_vec, 2); % Normalize our direction to the desired magnitude, 1. % Now generate a positive definite Toeplitz matrix as a deterministic function of our psuedorandom vector. for j = 1:n % Loop over the entries in the first row of the Toeplitz matrix. T_row(1, j) = phi_k(abs(1-j), n, rand_vec); % Produce the first row of the Toeplitz matrix. end rand_mat_mag = abs(randn(1, 1)); % Produce a random positive number by which to scale this matrix (just for some additional variation). T_row = rand_mat_mag*T_row; % Scale the first row by this random positive number. end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This is a subfunction which generates an element of a positive definite Toeplitz matrix, given an integer k = abs(i-j), the absolute value of the difference between the indices of the entry in the matrix, an integer n, the desired size of the matrix, and rand_vec, a n x 1 random vector in the unit n-sphere. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function[new_x] = phi_k(k, n, rand_vec) rand_vec(n+1:n+k+1, 1) = 0; % Extend our random vector with k more zeros. new_x = transpose(rand_vec(1:n, 1))*rand_vec(1+k:n+k, 1); % Compute the required sum in a vectorized way (from page 4 of "On random correlation matrices: II. The Toeplitz case" (1989) by R. B. Holmes). endPK C==%toeblitz-1.0/util/multiplyCirculant.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This uses a fast Fourier transform to compute the product y = Cx, where C is a circulant matrix and x is a column vector of appropriate size. This is a simplified version of the Toeplitz matrix multiplication on page 202 of "Matrix Computations", 3rd edition by Gene H. Golub. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [y] = multiplyCirculant(C, x) n = size(C, 2); %Gets the dimension of the circulant matrix. embed_circulant = C(1:end, 1); %Extracts information from first column of circulant matrix. y = ifft(fft(embed_circulant).*fft(x)); %Computes the result of the circulant matrix multiplication in a fast way. endPK Cc$(toeblitz-1.0/util/multiplyinvCirculant.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This function solves the matrix equation C*x = b for x, where C is a circulant matrix. This is equivalent to left-multiplying b by the inverse of C, C^(-1). It takes c, the first column of c, as input. C is never explicitly required or stored. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [x] = multiplyinvCirculant(c, b) x = ifft(fft(b)./fft(c)); % Performs the computation. The theoretical justification of this is a consequence of the fast Toeplitz multiplication method on page 202 of "Matrix Computations", 3rd edition by Gene H. Golub, and the properties of the Fourier transform. endPK C ;;$toeblitz-1.0/util/multiplyToeplitz.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This uses a fast Fourier transform to compute the product y = Tx, where T is a toeplitz matrix and x is a column vector of appropriate size. T is embedded in a circulant matrix and manipulated according to page 202 of "Matrix Computations", 3rd edition by Gene H. Golub. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [y] = multiplyToeplitz(T_row, x) n = size(T_row, 2); % Gets the dimension of the Toeplitz matrix. embed_circulant = T_row(1, end:-1:2); % Extracts information from first row of Toeplitz matrix. embed_circulant = vertcat(transpose(T_row(1, 1:end)), embed_circulant(:)); % Combines information from first column of Toeplitz matrix with the information from the first row extracted in the previous step. Note that these two steps can be condensed into a single faster step if the toeplitz matrix is known to be symmetric. y = ifft(fft(embed_circulant).*fft(vertcat(x, zeros(n-1, 1)))); % Computes the result of the circulant matrix multiplication in a fast way. y = y(1:n, 1); % Discards the extra trailing elements in the circulant matrix result, giving us the desired product of a Toeplitz matrix and a vector. endPK . C(oo)toeblitz-1.0/util/recreateToeblitzPlots.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang, 2013 % This function makes it convenient to reproduce all the plots included in the accompanying journal article at once. It's essentially a modified version of testToeplitz with no input arguments. You can just call it with no input variables and let it run. runMode is an optional variable, which if set to -1, will run the code for smaller matrices, making it easy to verify the code in a couple minutes, and allows one to run the code many times to find errors. If runMode is set to 0, it doesn't do anything, and the resulting behavior is equivalent to omitting it to begin with. Setting runMode to 1, 2, or 3 will run only a particular portion of the code needed to create one particular graph. A second optional input argument, fig_save, specifies whether we save the produced plots as MATLAB .fig files in addition to .pdf files. In order to specify the second input argument, runMode must be explicitly specified. fig_save defaults to 0 (only saves .pdf files) if not specified. % % In total, this function creates four plots when called in runMode 0 (default) or -1: one accuracy plot on a log-linear axis; one plot of iterations on a linear axis, and two loglog plots for complexity, one on a true loglog axis, and the other a linear plot of the logs of the two variables, used for regressions. When called in runMode 1, only the one accuracy plot is produced; when called in runMode 2, only the one iterations plot is produced; and when called in runMode 3, only the two complexity plots are produced. % % Calling syntax: % % recreateToeblitzPlots This simply runs the full recreateToeblitzPlots procedure. It should complete running overnight on a modern laptop computer. % recreateToeblitzPlots() This is equivalent to simply calling recreateToeblitzPlots. % recreateToeblitzPlots(0) This is equivalent to simply calling recreateToeblitzPlots. % recreateToeblitzPlots(-1) This runs a shorter version of the code to produce graphs for quick verification. % recreateToeblitzPlots(1) This produces only the accuracy check (full version) plot. % recreateToeblitzPlots(2) This produces only the iteration check (full version) for the various preconditioners. % recreateToeblitzPlots(3) This produces only the complexity testing (full version) plot for the various methods. % recreateToeblitzPlots(0, 0) This is equivalent to calling recreateToeblitzPlots(0) or simply recreateToeblitzPlots. % recreateToeblitzPlots(0, 1) This saves the produced figures as MATLAB .fig files in addition to .pdf files for editing and presentation. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [] = recreateToeblitzPlots(runMode, fig_save) % Set runMode to 0 by default. if nargin < 1 runMode = 0; end % Set fig_save to 0 by default. if nargin < 2 fig_save = 0; end % The following code sets default values for runs and save_flag if they are not specified. save_flag = 1; % Set save_flag to true. runs = 10; % Set the number of runs to 10. method = 'Placeholder'; % Set a placeholder for method at first. % For testing only: if runMode is -1 (quick verification), then only do a single run for each case, and do not save the output; instead, display it immediately. if runMode == -1 runs = 3; save_flag = 0; end % The following code finds a folder to save the result plots in, if save_flag is set to 1. folder = '0'; if save_flag for i = 1:999 % Searches from 001 to 999 for an unused folder name of the form 'test_XXX', where XXX is the three-digit number. if folder == '0' % Runs the following code if we have not yet found a suitable folder. test_folder = sprintf('./results/test_%03d/', i); % Finds the subdirectory in which the files will be placed. if not(exist(test_folder)) % Checks to see if there is already a folder with that name in the ./results/ directory. If not, it creates the folder and proceeds. folder = test_folder; % Changes the value of folder, stopping future checks for folder names. mkdir(folder); % Actually creates the directory. fprintf(['Printing to directory ' folder '.\n']) % Informs the user where the output will be saved. end end end end if or(or(runMode == 0, runMode == -1), runMode == 1) method = 'accuracyToeplitz'; % Sets the method to traceToeplitz. sizes_vec = [2:100:4502]; % Sets up a sizes_vec for small matrices. This will let us do a quick accuracy check. end % For testing only: if runMode == -1 sizes_vec = [100:20:200]; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The following code tests the accuracy of our methods. Routines for testing other functions work similarly. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if strcmp(method, 'accuracyToeplitz') % This checks if 'accuracyToeplitz' is the method specified for testing. accuracy_strs = cell(4, 1); % This creates a cell array of the names of our different methods for use in labeling plots and legends. accuracy_strs(1, 1) = cellstr('solveToeplitz'); accuracy_strs(2, 1) = cellstr('logdetToeplitz'); accuracy_strs(3, 1) = cellstr('invToeplitz'); accuracy_strs(4, 1) = cellstr('traceToeplitz'); vals_times = zeros(4, length(sizes_vec)); % This initializes the matrix in which all our test results will be (temporarily) stored. for i = 1:length(sizes_vec) % This loops over each the sizes of matrices we want to test. current_size_str = num2str(sizes_vec(i)); % Finds the current matrix size being worked on. new_vals_times = zeros(4, 1); % This initializes new_vals_times (and resets it for each size of matrix tested). for j = 1:runs % For each size matrix, this loops 'runs' times to obtain an average result. n = sizes_vec(i); run_str = num2str(j); % Finds the current run being worked on. fprintf(['Now on run number ' run_str ' of matrix size ' current_size_str ' for method ' method '.\n']) % Informs the user where we are in the program. T_row = genPDToeplitz(n); % This produces the first row of a (more or less) random positive definite Toeplitz matrix. b = transpose([1:n]); A = randn(n, n); new_vals_times = new_vals_times + test_accuracyToeplitz(T_row, A, b); % This calculates the trace of the product with an arbitrary matrix in various ways, and adds the timings and accuracy metrics to a cumulative total in new_vals_times (see test_traceToeplitz documentation for more detail). end vals_times(1:4, i) = new_vals_times(1:4, 1)/runs; % This saves the results in new_vals_times to the appropriate column in vals_times, dividing by the number of runs to obtain an average. end % Show or save results: These are results for accuracy. test_plot('semilogy', sizes_vec, vals_times(1:1:4, 1:end), accuracy_strs, 'Norm difference vs. built-in MATLAB methods', save_flag, folder, 'test_accuracy', fig_save); fprintf(['Printing to directory ' folder '.\n']) % Informs the user where the output will be saved. end if or(or(runMode == 0, runMode == -1), runMode == 2) method = 'iterToeplitz'; sizes_vec = [500:100:4500]; end % For testing only: if runMode == -1 sizes_vec = [50:50:250]; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The following code tests the number of iterations required for various preconditioned conjugate gradient methods. Routines for testing other functions work similarly. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if strcmp(method, 'iterToeplitz') % This checks if 'iterToeplitz' is the method specified for testing. iter_strs = cell(3, 1); % This creates a cell array of the names of our different methods for use in labeling plots and legends. iter_strs(1, 1) = cellstr('Unconditioned'); iter_strs(2, 1) = cellstr('Chan'); iter_strs(3, 1) = cellstr('Circulant Embedding'); vals_times = zeros(3, length(sizes_vec)); % This initializes the matrix in which all our test results will be (temporarily) stored. for i = 1:length(sizes_vec) % This loops over each the sizes of matrices we want to test. current_size_str = num2str(sizes_vec(i)); % Finds the current matrix size being worked on. new_vals_times = zeros(3, 1); % This initializes new_vals_times (and resets it for each size of matrix tested). for j = 1:runs % For each size matrix, this loops 'runs' times to obtain an average result. n = sizes_vec(i); run_str = num2str(j); % Finds the current run being worked on. fprintf(['Now on run number ' run_str ' of matrix size ' current_size_str ' for method ' method '.\n']) % Informs the user where we are in the program. T_row = genPDToeplitz(n); % This produces the first row of a (more or less) random positive definite Toeplitz matrix. b = transpose([1:n]); A = randn(n, n); new_vals_times = new_vals_times + test_iterToeplitz(T_row, b); % This calculates the trace of the product with an arbitrary matrix in various ways, and adds the timings and accuracy metrics to a cumulative total in new_vals_times (see test_traceToeplitz documentation for more detail). end vals_times(1:3, i) = new_vals_times(1:3, 1)/runs; % This saves the results in new_vals_times to the appropriate column in vals_times, dividing by the number of runs to obtain an average. end % Show or save results: These are results for accuracy. test_plot('linear', sizes_vec, vals_times(1:1:3, 1:end), iter_strs, 'Number of PCG Iterations', save_flag, folder, 'test_iter', fig_save); fprintf(['Printing to directory ' folder '.\n']) % Informs the user where the output will be saved. end if or(or(runMode == 0, runMode == -1), runMode == 3) method = 'complexityToeplitz'; % Sets the method to speedToeplitz. sizes_vec_small = floor(10.^[2.7:0.1:3.9]); % Sets up a sizes_vec_small for matrices large enough so that the computational time is measureable, but not so large that they break MATLAB. This will let us do a quick accuracy check. sizes_vec_large = floor(10.^[4:0.1:5]); % Sets up a sizes_vec_large for matrices large enough so that the computational time is measureable, but not so large that they break MATLAB. end % For testing only: if runMode == -1 sizes_vec_small = floor(10.^[2.7:0.2:3.5]); sizes_vec_large = floor(10.^[3.9:0.2:4.3]); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The following code tests the speed of our methods. Routines for testing other functions work similarly. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if strcmp(method, 'complexityToeplitz') % This checks if 'complexityToeplitz' is the method specified for testing. complexity_strs = cell(8, 1); % This creates a cell array of the names of our different methods for use in labeling plots and legends. complexity_strs(1, 1) = cellstr('solveToeplitz'); complexity_strs(2, 1) = cellstr('logdetToeplitz'); complexity_strs(3, 1) = cellstr('invToeplitz'); complexity_strs(4, 1) = cellstr('traceToeplitz'); complexity_strs(5, 1) = cellstr('MATLAB mldivide'); complexity_strs(6, 1) = cellstr('MATLAB logdet'); complexity_strs(7, 1) = cellstr('MATLAB inv'); complexity_strs(8, 1) = cellstr('MATLAB trace'); vals_times_small = zeros(8, length(sizes_vec_small)); % This initializes the matrix in which our test results for smaller matrices will be (temporarily) stored. vals_times_large = zeros(2, length(sizes_vec_large)); % This initializes the matrix in which our test results for larger matrices will be (temporarily) stored. for i = 1:length(sizes_vec_small) % This loops over each the sizes of matrices we want to test. n = sizes_vec_small(i); current_size_str = num2str(n); % Finds the current matrix size being worked on. new_vals_times = zeros(8, 1); % This initializes new_vals_times (and resets it for each size of matrix tested). for j = 1:runs % For each size matrix, this loops 'runs' times to obtain an average result. run_str = num2str(j); % Finds the current run being worked on. fprintf(['Now on run number ' run_str ' of matrix size ' current_size_str ' for method ' method '.\n']) % Informs the user where we are in the program. T_row = genPDToeplitz(n); % This produces the first row of a (more or less) random positive definite Toeplitz matrix. b = transpose([1:n]); A = randn(n, n); new_vals_times = new_vals_times + test_complexityToeplitz_small(T_row, A, b); % This calculates the trace of the product with an arbitrary matrix in various ways, and adds the timings and accuracy metrics to a cumulative total in new_vals_times (see test_traceToeplitz documentation for more detail). end vals_times_small(1:8, i) = new_vals_times(1:8, 1)/runs; % This saves the results in new_vals_times to the appropriate column in vals_times, dividing by the number of runs to obtain an average. end for i = 1:length(sizes_vec_large) % This loops over each the sizes of matrices we want to test. n = sizes_vec_large(i); current_size_str = num2str(n); % Finds the current matrix size being worked on. new_vals_times = zeros(2, 1); % This initializes new_vals_times (and resets it for each size of matrix tested). for j = 1:runs % For each size matrix, this loops 'runs' times to obtain an average result. run_str = num2str(j); % Finds the current run being worked on. fprintf(['Now on run number ' run_str ' of matrix size ' current_size_str ' for method ' method '.\n']) % Informs the user where we are in the program. T_row = genPDToeplitz(n); % This produces the first row of a (more or less) random positive definite Toeplitz matrix. b = transpose([1:n]); new_vals_times = new_vals_times + test_complexityToeplitz_large(T_row, b); % This calculates the trace of the product with an arbitrary matrix in various ways, and adds the timings and accuracy metrics to a cumulative total in new_vals_times (see test_traceToeplitz documentation for more detail). end vals_times_large(1:2, i) = new_vals_times(1:2, 1)/runs; % This saves the results in new_vals_times to the appropriate column in vals_times, dividing by the number of runs to obtain an average. end % Redistribute the data so that each actual data series is together in one row. vals_times_large = horzcat(vals_times_small(1:2, 1:end), vals_times_large); vals_times_small = vals_times_small(3:8, 1:end); % Show or save results: These are results for computational complexity. grand_loglogplot(sizes_vec_small, sizes_vec_large, vals_times_small, vals_times_large, complexity_strs, 'Computational Time', save_flag, folder, 'test_complexity', runMode, fig_save); grand_regressionplot(sizes_vec_small, sizes_vec_large, vals_times_small, vals_times_large, complexity_strs, 'Computational Time', save_flag, folder, 'test_complexityregression', runMode, fig_save); fprintf(['Printing to directory ' folder '.\n']) % Informs the user where the output will be saved. end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % test_accuracyToeplitz tests the accuracy of our Toeblitz method against built-in MATLAB functions and outputs the results in a column vector, new_vals_times. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function[new_vals_times] = test_accuracyToeplitz(T_row, A, b) new_vals_times = zeros(4, 1); % Intialize new_vals_times as a vector of zeros. % Check the accuracy of our preconditioned conjugate gradient method. x = solveToeplitzMldivide(T_row, b); % Get a standard against which to measure the accuracy of our Chan preconditioned conjugate gradient method. xPCG = solveToeplitzPCG(T_row, b, 'Chan'); % Get the result using our method. new_vals_times(1, 1) = norm(x(:) - xPCG(:), 2); % Compares accuracy to built-in Matlab algorithm. % Check the accuracy of the inverse in our decompToeplitz method. [TiInvT, ldT] = decompToeplitz(T_row, 0, -1); % Get a standard against which to measure the accuracy of our decompToeplitz method. [TiInvT0, ldT0] = decompToeplitz(T_row, 0, 0); % Compute the inverse and log determinant using our decompToeplitz method. new_vals_times(3, 1) = norm(TiInvT(:) - TiInvT0(:), 2); % Compares accuracy to built-in Matlab algorithm. TiInvT = 0; % Clears some bulky variables for performance. clearvars is not used due to Octave incompatbility. TiInvT0 = 0; % Clears some bulky variables for performance. clearvars is not used due to Octave incompatbility. % Check the accuracy of the log determinant in our decompToeplitz method. new_vals_times(2, 1) = norm(ldT(:) - ldT0(:), 2); % Compares accuracy to built-in Matlab algorithm. % Check the accuracy of our traceToeplitz method. tr = traceToeplitz(A, T_row, T_row, -1); % Get a standard against which to measure the accuracy of our traceToeplitz method. trC = traceToeplitz(A, T_row, T_row, 1); % Compute the trace using our method. new_vals_times(4, 1) = norm(tr - trC, 2); % Compares accuracy to built-in Matlab algorithm. end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % test_iterToeplitz tests the number of iterations required for the preconditioned conjugate gradient method to converge using various preconditioners, and outputs the results in a column vector, new_vals_times. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [new_vals_times] = test_iterToeplitz(T_row, b) new_vals_times = zeros(3, 1); % Intialize new_vals_times as a vector of zeros. % Call solveToeplitzPCG and specify the unconditioned conjugate gradient method. [xPCG, k] = solveToeplitzPCG(T_row, b, 'None'); new_vals_times(1, 1) = k; % Stores the number of iterations required to converge. % Call solveToeplitzPCG and specify the Chan preconditioned conjugate gradient method. [xPCG, k] = solveToeplitzPCG(T_row, b, 'Chan'); new_vals_times(2, 1) = k; % Stores the number of iterations required to converge. % Call solveToeplitzPCG and specify the Circulant Embedding preconditioned conjugate gradient method. t = cputime; [xPCG, k] = solveToeplitzPCG(T_row, b, 'Circulant_Embedding'); new_vals_times(3, 1) = k; % Stores the number of iterations required to converge. end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % test_complexityToeplitz_small tests the complexity of built-in MATLAB and Toeblitz methods limited by O(n^2) storage, and outputs the results in a column vector, new_vals_times. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function[new_vals_times] = test_complexityToeplitz_small(T_row, A, b); new_vals_times = zeros(8, 1); % Intialize new_vals_times as a vector of zeros. T = toeplitz(T_row); % Create the full matrix representation for built-in MATLAB methods. % Check the speed of our preconditioned conjugate gradient method. t = cputime; solveToeplitz(T_row, b); % Get the result using our method. new_vals_times(1, 1) = cputime - t; % Saves the time to our data matrix. % Check the speed of our logdetToeplitz method. t = cputime; logdetToeplitz(T_row); % Get the result using our method. new_vals_times(2, 1) = cputime - t; % Saves the time to our data matrix. % Check the speed of our invToeplitz method. t = cputime; invToeplitz(T_row); % Get the result using our method. new_vals_times(3, 1) = cputime - t; % Saves the time to our data matrix. % Check the speed of our traceToeplitz method. t = cputime; traceToeplitz(A, T_row, T_row, 1); % Compute the trace using our method. new_vals_times(4, 1) = cputime - t; % Saves the time to our data matrix. % Check the speed of the built-in matrix equation solver. t = cputime; mldivide(T, b); % Get the result using the built-in MATLAB method. new_vals_times(5, 1) = cputime - t; % Saves the time to our data matrix. % Check the speed of the built-in log determinant method. t = cputime; logdet(T); % Get the result using the built-in MATLAB method. new_vals_times(6, 1) = cputime - t; % Saves the time to our data matrix. % Check the speed of the built-in matrix inversion method. t = cputime; inv(T); % Get the result using the built-in MATLAB method. new_vals_times(7, 1) = cputime - t; % Saves the time to our data matrix. % Check the speed of the built-in matrix multiply and trace method. t = cputime; trace(A*T); % Get the result using the built-in MATLAB method. % sum(sum(A.*transpose(T))); % Get the result using the built-in MATLAB method. This line is actually a vast improvement over simply taking trace(A*T). new_vals_times(8, 1) = cputime - t; % Saves the time to our data matrix. end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % test_complexityToeplitz_large tests the complexity of Toeblitz methods that use O(n) storage, and outputs the results in a column vector, new_vals_times. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function[new_vals_times] = test_complexityToeplitz_large(T_row, b); new_vals_times = zeros(2, 1); % Intialize new_vals_times as a vector of zeros. % Check the speed of our preconditioned conjugate gradient method. t = cputime; solveToeplitz(T_row, b); % Get the result using our method. new_vals_times(1, 1) = cputime - t; % Saves the time to our data matrix. % Check the speed of our logdetToeplitz method. t = cputime; logdetToeplitz(T_row); % Compute just the log determinant using our logdetToeplitz method. new_vals_times(2, 1) = cputime - t; % Saves the time to our data matrix. end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % grand_loglogplot plots the computational time used for calculations vs. matrix size for several methods on a log-log axis, allowing us to compare the complexities of the various methods. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [a_graph] = grand_loglogplot(sizes_vec_small, sizes_vec_large, data_mat_small, data_mat_large, data_labels, yaxis_label, save_flag, folder, file_name, runMode, fig_save) sizes_vec_large = horzcat(sizes_vec_small, sizes_vec_large); % Adds sizes_vec_small to the beginning of sizes_vec_large so that the full matrix size series in present. a_graph = figure; % Create a figure. loglog(sizes_vec_large, data_mat_large(1, 1:end), 'r', sizes_vec_large, data_mat_large(2, 1:end), 'b', sizes_vec_small, data_mat_small(1, 1:end), 'm', sizes_vec_small, data_mat_small(2, 1:end), 'g', sizes_vec_small, data_mat_small(3, 1:end), 'r--', sizes_vec_small, data_mat_small(4, 1:end), 'b--', sizes_vec_small, data_mat_small(5, 1:end), 'm--', sizes_vec_small, data_mat_small(6, 1:end), 'g--', 'LineWidth',3); % Plot our data. xlabel('Matrix Size', 'FontSize', 18); % Label the x-axis. ylabel(yaxis_label,'FontSize', 18); % Label the y-axis. set(gca,'fontsize', 18); % Set the general font size. legend(char(data_labels(1, 1)), char(data_labels(2, 1)), char(data_labels(3, 1)), char(data_labels(4, 1)), char(data_labels(5, 1)), char(data_labels(6, 1)), char(data_labels(7, 1)), char(data_labels(8, 1))) % Label the legend using the provided strings. % Save our plot if save_flag is on. Otherwise, display it. if save_flag print(a_graph, strcat(folder, file_name), '-dpdf'); if fig_save print(a_graph, strcat(folder, file_name), '-dfig'); end close all else a_graph; end %{ % View basic semilogy and linear plots of our data when in runMode == -1. This helps us see what's actually being plotted, since loglog has trouble with values at or near 0. Use for debugging. if runMode == -1 figure; semilogy(sizes_vec_large, data_mat_large(1, 1:end), 'r', sizes_vec_large, data_mat_large(2, 1:end), 'b', sizes_vec_small, data_mat_small(1, 1:end), 'm', sizes_vec_small, data_mat_small(2, 1:end), 'g', sizes_vec_small, data_mat_small(3, 1:end), 'r--', sizes_vec_small, data_mat_small(4, 1:end), 'b--', sizes_vec_small, data_mat_small(5, 1:end), 'm--', sizes_vec_small, data_mat_small(6, 1:end), 'g--', 'LineWidth',3); figure; plot(sizes_vec_large, data_mat_large(1, 1:end), 'r', sizes_vec_large, data_mat_large(2, 1:end), 'b', sizes_vec_small, data_mat_small(1, 1:end), 'm', sizes_vec_small, data_mat_small(2, 1:end), 'g', sizes_vec_small, data_mat_small(3, 1:end), 'r--', sizes_vec_small, data_mat_small(4, 1:end), 'b--', sizes_vec_small, data_mat_small(5, 1:end), 'm--', sizes_vec_small, data_mat_small(6, 1:end), 'g--', 'LineWidth',3); end %} end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % grand_regressionplot plots the log of the computational time used for calculations vs. the log of the matrix size for several methods on a linear axis, effectively producing the same figure as grand_loglogplot. However, this format allows us to use MATLAB's simple regression tools to determine the best fit slopes for each data series obtained. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [a_graph] = grand_regressionplot(sizes_vec_small, sizes_vec_large, data_mat_small, data_mat_large, data_labels, yaxis_label, save_flag, folder, file_name, runMode, fig_save) sizes_vec_large = horzcat(sizes_vec_small, sizes_vec_large); % Adds sizes_vec_small to the beginning of sizes_vec_large so that the full matrix size series in present. % Get the log of all the data, so that a normal plot will look the same as a loglog plot. sizes_vec_large = log(sizes_vec_large); sizes_vec_small = log(sizes_vec_small); data_mat_large = log(data_mat_large); data_mat_small = log(data_mat_small); a_graph = figure; % Create a figure. plot(sizes_vec_large, data_mat_large(1, 1:end), 'r', sizes_vec_large, data_mat_large(2, 1:end), 'b', sizes_vec_small, data_mat_small(1, 1:end), 'm', sizes_vec_small, data_mat_small(2, 1:end), 'g', sizes_vec_small, data_mat_small(3, 1:end), 'r--', sizes_vec_small, data_mat_small(4, 1:end), 'b--', sizes_vec_small, data_mat_small(5, 1:end), 'm--', sizes_vec_small, data_mat_small(6, 1:end), 'g--', 'LineWidth',3); % Plot our data. xlabel('Matrix Size', 'FontSize', 18); % Label the x-axis. ylabel(yaxis_label,'FontSize', 18); % Label the y-axis. set(gca,'fontsize', 18); % Set the general font size. legend(char(data_labels(1, 1)), char(data_labels(2, 1)), char(data_labels(3, 1)), char(data_labels(4, 1)), char(data_labels(5, 1)), char(data_labels(6, 1)), char(data_labels(7, 1)), char(data_labels(8, 1))) % Label the legend using the provided strings. % Save our plot if save_flag is on. Otherwise, display it. if save_flag print(a_graph, strcat(folder, file_name), '-dpdf'); if fig_save print(a_graph, strcat(folder, file_name), '-dfig'); end close all else a_graph; end %{ % View basic semilogy and linear plots of our data when in runMode. This helps us see what's actually being plotted, since loglog has trouble with values at or near 0. if runMode == -1 figure; semilogy(sizes_vec_large, data_mat_large(1, 1:end), 'r--', sizes_vec_large, data_mat_large(2, 1:end), 'g--', sizes_vec_small, data_mat_small(1, 1:end), 'b--', sizes_vec_small, data_mat_small(2, 1:end), 'y--', sizes_vec_small, data_mat_small(3, 1:end), 'r', sizes_vec_small, data_mat_small(4, 1:end), 'g', sizes_vec_small, data_mat_small(5, 1:end), 'b', sizes_vec_small, data_mat_small(6, 1:end), 'y', 'LineWidth',3); figure; plot(sizes_vec_large, data_mat_large(1, 1:end), 'r--', sizes_vec_large, data_mat_large(2, 1:end), 'g--', sizes_vec_small, data_mat_small(1, 1:end), 'b--', sizes_vec_small, data_mat_small(2, 1:end), 'y--', sizes_vec_small, data_mat_small(3, 1:end), 'r', sizes_vec_small, data_mat_small(4, 1:end), 'g', sizes_vec_small, data_mat_small(5, 1:end), 'b', sizes_vec_small, data_mat_small(6, 1:end), 'y', 'LineWidth',3); end %} end PK C:& $toeblitz-1.0/util/startup_checkMEX.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Toeblitz: A Fast Toolkit for Manipulating Toeplitz Matrices % Copyright (C) 2013 William B. Zhang and John P. Cunningham (see full notice in README) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % William Zhang and John Cunningham, 2013 % The following code checks for the relevant MEX files (such as .mexa64 % or .mexglx, depending on the machine architecture), and it creates the % mex file if it can not find the right one. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [] = startup_checkMEX(file_directory, file_name) toeplitz_directory = sprintf('%s', pwd); % Finds the Toeplitz directory in a stable way. toeplitz_directory = strrep(toeplitz_directory, '\', '/'); % Replaces Windows backslashes with forward slashes. file_directory = [toeplitz_directory '/' file_directory]; % Creates the full file directory. if not(exist(sprintf([file_directory '/' file_name '.' mexext]), 'file')) % Check if the relevant files exist. try % If not, try to create them. if isOctave eval(sprintf(['mkoctfile --mex --output ' file_directory '/' file_name '.mex ' file_directory '/' file_name '.c'])) else eval(sprintf(['mex -outdir ' file_directory ' ' file_directory '/' file_name '.c'])); % This line actually compiles the C code and creates the MEX file. end fprintf(['NOTE: the relevant ' file_name ' mex files were not found. They have been created.\n']); % Success message. catch % If the files cannot be created, display the following error message. fprintf(['NOTE: the relevant ' file_name ' mex files were not found, and your machine failed to create them.\n']); fprintf(' This usually means that you do not have the proper C/MEX compiler setup.\n'); fprintf(' The code will still run identically, albeit slower (perhaps considerably).\n'); fprintf(' Please read the README file section on the use of C/MEX.\n'); end end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The function isOctave detects whether the user is using Octave, returning true if this is the case, and false if it is not (meaning the user is using MATLAB, most likely). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [OctaveMode] = isOctave() OctaveMode = exist('OCTAVE_VERSION') ~= 0; end PKu Etoeblitz-1.0/written/PK Eӥ<<!toeblitz-1.0/written/toeblitz.pdf%PDF-1.5 % 3 0 obj << /Length 3749 /Filter /FlateDecode >> stream x[۶BFD "MqƗ'')ϗX ;ON_N$X`},3_^>k-LՂ%:N8[F6ߢ˦+'*nbRI=YVȨk5՛WMK_돈YTjvy޷{~6˦v 96aNDM\VU`n&/,m^_&) DH͡뫸<>rWF4yT,ڔk+XY,꾴 Di..aOv4^(PNœiOuhJ7eDY q;?ARX~z$t"1f[D(RjuKBOpk2ź2?~b"V-boU;8myN -v :j7tȝdI$&q3gEO=MR3B*R8]E ̞|g}j]tteWe Tyh󪺥k=vz ?`iTGUVaf[;(ZǕȺj:~o.*ǿ#:3J'! 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