! This is a test program for checking the implementations of ! the implementations of the following subroutines ! ! DGEDMD for computation of the ! Dynamic Mode Decomposition (DMD) ! DGEDMDQ for computation of a ! QR factorization based compressed DMD ! ! Developed and supported by: ! =========================== ! Developed and coded by Zlatko Drmac, Faculty of Science, ! University of Zagreb; drmac@math.hr ! In cooperation with ! AIMdyn Inc., Santa Barbara, CA. ! ======================================================== ! How to run the code (compiler, link info) ! ======================================================== ! Compile as FORTRAN 90 (or later) and link with BLAS and ! LAPACK libraries. ! NOTE: The code is developed and tested on top of the ! Intel MKL library (versions 2022.0.3 and 2022.2.0), ! using the Intel Fortran compiler. ! ! For developers of the C++ implementation ! ======================================================== ! See the LAPACK++ and Template Numerical Toolkit (TNT) ! ! Note on a development of the GPU HP implementation ! ======================================================== ! Work in progress. See CUDA, MAGMA, SLATE. ! NOTE: The four SVD subroutines used in this code are ! included as a part of R&D and for the completeness. ! This was also an opportunity to test those SVD codes. ! If the scaling option is used all four are essentially ! equally good. For implementations on HP platforms, ! one can use whichever SVD is available. !... ......................................................... ! NOTE: ! When using the Intel MKL 2022.0.3 the subroutine xGESVDQ ! (optionally used in xGEDMD) may cause access violation ! error for x = S, D, C, Z, but only if called with the ! work space query. (At least in our Windows 10 MSVS 2019.) ! The problem can be mitigated by downloading the source ! code of xGESVDQ from the LAPACK repository and use it ! localy instead of the one in the MKL. This seems to ! indicate that the problem is indeed in the MKL. ! This problem did not appear whith Intel MKL 2022.2.0. ! ! NOTE: ! xGESDD seems to have a problem with workspace. In some ! cases the length of the optimal workspace is returned ! smaller than the minimal workspace, as specified in the ! code. As a precaution, all optimal workspaces are ! set as MAX(minimal, optimal). ! Latest implementations of complex xGESDD have different ! length of the real worksapce. We use max value over ! two versions. !............................................................ !............................................................ ! PROGRAM DMD_TEST use iso_fortran_env, only: real64 IMPLICIT NONE integer, parameter :: WP = real64 !............................................................ REAL(KIND=WP), PARAMETER :: ONE = 1.0_WP REAL(KIND=WP), PARAMETER :: ZERO = 0.0_WP !............................................................ REAL(KIND=WP), ALLOCATABLE, DIMENSION(:,:) :: & A, AC, EIGA, LAMBDA, LAMBDAQ, F, F1, F2,& Z, Z1, S, AU, W, VA, X, X0, Y, Y0, Y1 REAL(KIND=WP), ALLOCATABLE, DIMENSION(:) :: & DA, DL, DR, REIG, REIGA, REIGQ, IEIG, & IEIGA, IEIGQ, RES, RES1, RESEX, SINGVX,& SINGVQX, WORK INTEGER , ALLOCATABLE, DIMENSION(:) :: IWORK REAL(KIND=WP) :: AB(2,2), WDUMMY(2) INTEGER :: IDUMMY(2), ISEED(4), RJOBDATA(8) REAL(KIND=WP) :: ANORM, COND, CONDL, CONDR, DMAX, EPS, & TOL, TOL2, SVDIFF, TMP, TMP_AU, & TMP_FQR, TMP_REZ, TMP_REZQ, TMP_ZXW, & TMP_EX, XNORM, YNORM !............................................................ INTEGER :: K, KQ, LDF, LDS, LDA, LDAU, LDW, LDX, LDY, & LDZ, LIWORK, LWORK, M, N, L, LLOOP, NRNK INTEGER :: i, iJOBREF, iJOBZ, iSCALE, INFO, j, KDIFF, & NFAIL, NFAIL_AU, NFAIL_F_QR, NFAIL_REZ, & NFAIL_REZQ, NFAIL_SVDIFF, NFAIL_TOTAL, NFAILQ_TOTAL, & NFAIL_Z_XV, MODE, MODEL, MODER, WHTSVD INTEGER iNRNK, iWHTSVD, K_TRAJ, LWMINOPT CHARACTER(LEN=1) GRADE, JOBREF, JOBZ, PIVTNG, RSIGN, & SCALE, RESIDS, WANTQ, WANTR LOGICAL TEST_QRDMD !..... external subroutines (BLAS and LAPACK) EXTERNAL DAXPY, DGEEV, DGEMM, DGEMV, DLACPY, DLASCL EXTERNAL DLARNV, DLATMR !.....external subroutines DMD package, part 1 ! subroutines under test EXTERNAL DGEDMD, DGEDMDQ !..... external functions (BLAS and LAPACK) EXTERNAL DLAMCH, DLANGE, DNRM2 REAL(KIND=WP) :: DLAMCH, DLANGE, DNRM2 EXTERNAL LSAME LOGICAL LSAME INTRINSIC ABS, INT, MIN, MAX !............................................................ ! The test is always in pairs : ( DGEDMD and DGEDMDQ ) ! because the test includes comparing the results (in pairs). !..................................................................................... TEST_QRDMD = .TRUE. ! This code by default performs tests on DGEDMDQ ! Since the QR factorizations based algorithm is designed for ! single trajectory data, only single trajectory tests will ! be performed with xGEDMDQ. WANTQ = 'Q' WANTR = 'R' !................................................................................. EPS = DLAMCH( 'P' ) ! machine precision DP ! Global counters of failures of some particular tests NFAIL = 0 NFAIL_REZ = 0 NFAIL_REZQ = 0 NFAIL_Z_XV = 0 NFAIL_F_QR = 0 NFAIL_AU = 0 KDIFF = 0 NFAIL_SVDIFF = 0 NFAIL_TOTAL = 0 NFAILQ_TOTAL = 0 DO LLOOP = 1, 4 WRITE(*,*) 'L Loop Index = ', LLOOP ! Set the dimensions of the problem ... WRITE(*,*) 'M = ' READ(*,*) M WRITE(*,*) M ! ... and the number of snapshots. WRITE(*,*) 'N = ' READ(*,*) N WRITE(*,*) N ! ... Test the dimensions IF ( ( MIN(M,N) == 0 ) .OR. ( M < N ) ) THEN WRITE(*,*) 'Bad dimensions. Required: M >= N > 0.' STOP END IF !............. ! The seed inside the LLOOP so that each pass can be reproduced easily. ISEED(1) = 4 ISEED(2) = 3 ISEED(3) = 2 ISEED(4) = 1 LDA = M LDF = M LDX = MAX(M,N+1) LDY = MAX(M,N+1) LDW = N LDZ = M LDAU = MAX(M,N+1) LDS = N TMP_ZXW = ZERO TMP_AU = ZERO TMP_REZ = ZERO TMP_REZQ = ZERO SVDIFF = ZERO TMP_EX = ZERO ! ! Test the subroutines on real data snapshots. All ! computation is done in real arithmetic, even when ! Koopman eigenvalues and modes are real. ! ! Allocate memory space ALLOCATE( A(LDA,M) ) ALLOCATE( AC(LDA,M) ) ALLOCATE( DA(M) ) ALLOCATE( DL(M) ) ALLOCATE( F(LDF,N+1) ) ALLOCATE( F1(LDF,N+1) ) ALLOCATE( F2(LDF,N+1) ) ALLOCATE( X(LDX,N) ) ALLOCATE( X0(LDX,N) ) ALLOCATE( SINGVX(N) ) ALLOCATE( SINGVQX(N) ) ALLOCATE( Y(LDY,N+1) ) ALLOCATE( Y0(LDY,N+1) ) ALLOCATE( Y1(M,N+1) ) ALLOCATE( Z(LDZ,N) ) ALLOCATE( Z1(LDZ,N) ) ALLOCATE( RES(N) ) ALLOCATE( RES1(N) ) ALLOCATE( RESEX(N) ) ALLOCATE( REIG(N) ) ALLOCATE( IEIG(N) ) ALLOCATE( REIGQ(N) ) ALLOCATE( IEIGQ(N) ) ALLOCATE( REIGA(M) ) ALLOCATE( IEIGA(M) ) ALLOCATE( VA(LDA,M) ) ALLOCATE( LAMBDA(N,2) ) ALLOCATE( LAMBDAQ(N,2) ) ALLOCATE( EIGA(M,2) ) ALLOCATE( W(LDW,N) ) ALLOCATE( AU(LDAU,N) ) ALLOCATE( S(N,N) ) TOL = M*EPS ! This mimics O(M*N)*EPS bound for accumulated roundoff error. ! The factor 10 is somewhat arbitrary. TOL2 = 10*M*N*EPS !............. DO K_TRAJ = 1, 2 ! Number of intial conditions in the simulation/trajectories (1 or 2) COND = 1.0D8 DMAX = 1.0D2 RSIGN = 'F' GRADE = 'N' MODEL = 6 CONDL = 1.0D2 MODER = 6 CONDR = 1.0D2 PIVTNG = 'N' ! Loop over all parameter MODE values for ZLATMR (+1,..,+6) DO MODE = 1, 6 ALLOCATE( IWORK(2*M) ) ALLOCATE(DR(N)) CALL DLATMR( M, M, 'S', ISEED, 'N', DA, MODE, COND, & DMAX, RSIGN, GRADE, DL, MODEL, CONDL, & DR, MODER, CONDR, PIVTNG, IWORK, M, M, & ZERO, -ONE, 'N', A, LDA, IWORK(M+1), INFO ) DEALLOCATE(IWORK) DEALLOCATE(DR) LWORK = 4*M+1 ALLOCATE(WORK(LWORK)) AC = A CALL DGEEV( 'N','V', M, AC, M, REIGA, IEIGA, VA, M, & VA, M, WORK, LWORK, INFO ) ! LAPACK CALL DEALLOCATE(WORK) TMP = ZERO DO i = 1, M EIGA(i,1) = REIGA(i) EIGA(i,2) = IEIGA(i) TMP = MAX( TMP, SQRT(REIGA(i)**2+IEIGA(i)**2)) END DO ! Scale A to have the desirable spectral radius. CALL DLASCL( 'G', 0, 0, TMP, ONE, M, M, A, M, INFO ) CALL DLASCL( 'G', 0, 0, TMP, ONE, M, 2, EIGA, M, INFO ) ! Compute the norm of A ANORM = DLANGE( 'F', N, N, A, M, WDUMMY ) IF ( K_TRAJ == 2 ) THEN ! generate data with two inital conditions CALL DLARNV(2, ISEED, M, F1(1,1) ) F1(1:M,1) = 1.0E-10*F1(1:M,1) DO i = 1, N/2 CALL DGEMV( 'N', M, M, ONE, A, M, F1(1,i), 1, ZERO, & F1(1,i+1), 1 ) END DO X0(1:M,1:N/2) = F1(1:M,1:N/2) Y0(1:M,1:N/2) = F1(1:M,2:N/2+1) CALL DLARNV(2, ISEED, M, F1(1,1) ) DO i = 1, N-N/2 CALL DGEMV( 'N', M, M, ONE, A, M, F1(1,i), 1, ZERO, & F1(1,i+1), 1 ) END DO X0(1:M,N/2+1:N) = F1(1:M,1:N-N/2) Y0(1:M,N/2+1:N) = F1(1:M,2:N-N/2+1) ELSE CALL DLARNV(2, ISEED, M, F(1,1) ) DO i = 1, N CALL DGEMV( 'N', M, M, ONE, A, M, F(1,i), 1, ZERO, & F(1,i+1), 1 ) END DO X0(1:M,1:N) = F(1:M,1:N) Y0(1:M,1:N) = F(1:M,2:N+1) END IF XNORM = DLANGE( 'F', M, N, X0, LDX, WDUMMY ) YNORM = DLANGE( 'F', M, N, Y0, LDX, WDUMMY ) !............................................................ DO iJOBZ = 1, 4 SELECT CASE ( iJOBZ ) CASE(1) JOBZ = 'V' ! Ritz vectors will be computed RESIDS = 'R' ! Residuals will be computed CASE(2) JOBZ = 'V' RESIDS = 'N' CASE(3) JOBZ = 'F' ! Ritz vectors in factored form RESIDS = 'N' CASE(4) JOBZ = 'N' RESIDS = 'N' END SELECT DO iJOBREF = 1, 3 SELECT CASE ( iJOBREF ) CASE(1) JOBREF = 'R' ! Data for refined Ritz vectors CASE(2) JOBREF = 'E' ! Exact DMD vectors CASE(3) JOBREF = 'N' END SELECT DO iSCALE = 1, 4 SELECT CASE ( iSCALE ) CASE(1) SCALE = 'S' ! X data normalized CASE(2) SCALE = 'C' ! X normalized, consist. check CASE(3) SCALE = 'Y' ! Y data normalized CASE(4) SCALE = 'N' END SELECT DO iNRNK = -1, -2, -1 ! Two truncation strategies. The "-2" case for R&D ! purposes only - it uses possibly low accuracy small ! singular values, in which case the formulas used in ! the DMD are highly sensitive. NRNK = iNRNK DO iWHTSVD = 1, 4 ! Check all four options to compute the POD basis ! via the SVD. WHTSVD = iWHTSVD DO LWMINOPT = 1, 2 ! Workspace query for the minimal (1) and for the optimal ! (2) workspace lengths determined by workspace query. X(1:M,1:N) = X0(1:M,1:N) Y(1:M,1:N) = Y0(1:M,1:N) ! DGEDMD: Workspace query and workspace allocation CALL DGEDMD( SCALE, JOBZ, RESIDS, JOBREF, WHTSVD, M, & N, X, LDX, Y, LDY, NRNK, TOL, K, REIG, IEIG, Z, & LDZ, RES, AU, LDAU, W, LDW, S, LDS, WDUMMY, -1, & IDUMMY, -1, INFO ) LIWORK = IDUMMY(1) ALLOCATE( IWORK(LIWORK) ) LWORK = INT(WDUMMY(LWMINOPT)) ALLOCATE( WORK(LWORK) ) ! DGEDMD test: CALL DGEDMD CALL DGEDMD( SCALE, JOBZ, RESIDS, JOBREF, WHTSVD, M, & N, X, LDX, Y, LDY, NRNK, TOL, K, REIG, IEIG, Z, & LDZ, RES, AU, LDAU, W, LDW, S, LDS, WORK, LWORK,& IWORK, LIWORK, INFO ) SINGVX(1:N) = WORK(1:N) !...... DGEDMD check point IF ( LSAME(JOBZ,'V') ) THEN ! Check that Z = X*W, on return from DGEDMD ! This checks that the returned aigenvectors in Z are ! the product of the SVD'POD basis returned in X ! and the eigenvectors of the rayleigh quotient ! returned in W CALL DGEMM( 'N', 'N', M, K, K, ONE, X, LDX, W, LDW, & ZERO, Z1, LDZ ) TMP = ZERO DO i = 1, K CALL DAXPY( M, -ONE, Z(1,i), 1, Z1(1,i), 1) TMP = MAX(TMP, DNRM2( M, Z1(1,i), 1 ) ) END DO TMP_ZXW = MAX(TMP_ZXW, TMP ) IF ( TMP_ZXW > 10*M*EPS ) THEN NFAIL_Z_XV = NFAIL_Z_XV + 1 WRITE(*,*) ':( .................DGEDMD FAILED!', & 'Check the code for implementation errors.' WRITE(*,*) 'The input parameters were ',& SCALE, JOBZ, RESIDS, JOBREF, WHTSVD, & M, N, LDX, LDY, NRNK, TOL END IF END IF !...... DGEDMD check point IF ( LSAME(JOBREF,'R') ) THEN ! The matrix A*U is returned for computing refined Ritz vectors. ! Check that A*U is computed correctly using the formula ! A*U = Y * V * inv(SIGMA). This depends on the ! accuracy in the computed singular values and vectors of X. ! See the paper for an error analysis. ! Note that the left singular vectors of the input matrix X ! are returned in the array X. CALL DGEMM( 'N', 'N', M, K, M, ONE, A, LDA, X, LDX, & ZERO, Z1, LDZ ) TMP = ZERO DO i = 1, K CALL DAXPY( M, -ONE, AU(1,i), 1, Z1(1,i), 1) TMP = MAX( TMP, DNRM2( M, Z1(1,i),1 ) * & SINGVX(K)/(ANORM*SINGVX(1)) ) END DO TMP_AU = MAX( TMP_AU, TMP ) IF ( TMP > TOL2 ) THEN NFAIL_AU = NFAIL_AU + 1 WRITE(*,*) ':( .................DGEDMD FAILED!', & 'Check the code for implementation errors.' WRITE(*,*) 'The input parameters were ',& SCALE, JOBZ, RESIDS, JOBREF, WHTSVD, & M, N, LDX, LDY, NRNK, TOL END IF ELSEIF ( LSAME(JOBREF,'E') ) THEN ! The unscaled vectors of the Exact DMD are computed. ! This option is included for the sake of completeness, ! for users who prefer the Exact DMD vectors. The ! returned vectors are in the real form, in the same way ! as the Ritz vectors. Here we just save the vectors ! and test them separately using a Matlab script. CALL DGEMM( 'N', 'N', M, K, M, ONE, A, LDA, AU, LDAU, ZERO, Y1, M ) i=1 DO WHILE ( i <= K ) IF ( IEIG(i) == ZERO ) THEN ! have a real eigenvalue with real eigenvector CALL DAXPY( M, -REIG(i), AU(1,i), 1, Y1(1,i), 1 ) RESEX(i) = DNRM2( M, Y1(1,i), 1) / DNRM2(M,AU(1,i),1) i = i + 1 ELSE ! Have a complex conjugate pair ! REIG(i) +- sqrt(-1)*IMEIG(i). ! Since all computation is done in real ! arithmetic, the formula for the residual ! is recast for real representation of the ! complex conjugate eigenpair. See the ! description of RES. AB(1,1) = REIG(i) AB(2,1) = -IEIG(i) AB(1,2) = IEIG(i) AB(2,2) = REIG(i) CALL DGEMM( 'N', 'N', M, 2, 2, -ONE, AU(1,i), & M, AB, 2, ONE, Y1(1,i), M ) RESEX(i) = DLANGE( 'F', M, 2, Y1(1,i), M, & WORK )/ DLANGE( 'F', M, 2, AU(1,i), M, & WORK ) RESEX(i+1) = RESEX(i) i = i + 2 END IF END DO END IF !...... DGEDMD check point IF ( LSAME(RESIDS, 'R') ) THEN ! Compare the residuals returned by DGEDMD with the ! explicitly computed residuals using the matrix A. ! Compute explicitly Y1 = A*Z CALL DGEMM( 'N', 'N', M, K, M, ONE, A, LDA, Z, LDZ, ZERO, Y1, M ) ! ... and then A*Z(:,i) - LAMBDA(i)*Z(:,i), using the real forms ! of the invariant subspaces that correspond to complex conjugate ! pairs of eigencalues. (See the description of Z in DGEDMD,) i = 1 DO WHILE ( i <= K ) IF ( IEIG(i) == ZERO ) THEN ! have a real eigenvalue with real eigenvector CALL DAXPY( M, -REIG(i), Z(1,i), 1, Y1(1,i), 1 ) RES1(i) = DNRM2( M, Y1(1,i), 1) i = i + 1 ELSE ! Have a complex conjugate pair ! REIG(i) +- sqrt(-1)*IMEIG(i). ! Since all computation is done in real ! arithmetic, the formula for the residual ! is recast for real representation of the ! complex conjugate eigenpair. See the ! description of RES. AB(1,1) = REIG(i) AB(2,1) = -IEIG(i) AB(1,2) = IEIG(i) AB(2,2) = REIG(i) CALL DGEMM( 'N', 'N', M, 2, 2, -ONE, Z(1,i), & M, AB, 2, ONE, Y1(1,i), M ) RES1(i) = DLANGE( 'F', M, 2, Y1(1,i), M, & WORK ) RES1(i+1) = RES1(i) i = i + 2 END IF END DO TMP = ZERO DO i = 1, K TMP = MAX( TMP, ABS(RES(i) - RES1(i)) * & SINGVX(K)/(ANORM*SINGVX(1)) ) END DO TMP_REZ = MAX( TMP_REZ, TMP ) IF ( TMP > TOL2 ) THEN NFAIL_REZ = NFAIL_REZ + 1 WRITE(*,*) ':( ..................DGEDMD FAILED!', & 'Check the code for implementation errors.' WRITE(*,*) 'The input parameters were ',& SCALE, JOBZ, RESIDS, JOBREF, WHTSVD, & M, N, LDX, LDY, NRNK, TOL END IF IF ( LSAME(JOBREF,'E') ) THEN TMP = ZERO DO i = 1, K TMP = MAX( TMP, ABS(RES1(i) - RESEX(i))/(RES1(i)+RESEX(i)) ) END DO TMP_EX = MAX(TMP_EX,TMP) END IF END IF !..... store the results for inspection DO i = 1, K LAMBDA(i,1) = REIG(i) LAMBDA(i,2) = IEIG(i) END DO DEALLOCATE(IWORK) DEALLOCATE(WORK) !====================================================================== ! Now test the DGEDMDQ !====================================================================== IF ( TEST_QRDMD .AND. (K_TRAJ == 1) ) THEN RJOBDATA(2) = 1 F1 = F ! DGEDMDQ test: Workspace query and workspace allocation CALL DGEDMDQ( SCALE, JOBZ, RESIDS, WANTQ, WANTR, & JOBREF, WHTSVD, M, N+1, F1, LDF, X, LDX, Y, & LDY, NRNK, TOL, KQ, REIGQ, IEIGQ, Z, LDZ, & RES, AU, LDAU, W, LDW, S, LDS, WDUMMY, & -1, IDUMMY, -1, INFO ) LIWORK = IDUMMY(1) ALLOCATE( IWORK(LIWORK) ) LWORK = INT(WDUMMY(LWMINOPT)) ALLOCATE(WORK(LWORK)) ! DGEDMDQ test: CALL DGEDMDQ CALL DGEDMDQ( SCALE, JOBZ, RESIDS, WANTQ, WANTR, & JOBREF, WHTSVD, M, N+1, F1, LDF, X, LDX, Y, & LDY, NRNK, TOL, KQ, REIGQ, IEIGQ, Z, LDZ, & RES, AU, LDAU, W, LDW, S, LDS, & WORK, LWORK, IWORK, LIWORK, INFO ) SINGVQX(1:KQ) = WORK(MIN(M,N+1)+1: MIN(M,N+1)+KQ) !..... DGEDMDQ check point IF ( KQ /= K ) THEN KDIFF = KDIFF+1 END IF TMP = ZERO DO i = 1, MIN(K, KQ) TMP = MAX(TMP, ABS(SINGVX(i)-SINGVQX(i)) / & SINGVX(1) ) END DO SVDIFF = MAX( SVDIFF, TMP ) IF ( TMP > M*N*EPS ) THEN WRITE(*,*) 'FAILED! Something was wrong with the run.' NFAIL_SVDIFF = NFAIL_SVDIFF + 1 DO j =1, 3 write(*,*) j, SINGVX(j), SINGVQX(j) read(*,*) END DO END IF !..... DGEDMDQ check point IF ( LSAME(WANTQ,'Q') .AND. LSAME(WANTR,'R') ) THEN ! Check that the QR factors are computed and returned ! as requested. The residual ||F-Q*R||_F / ||F||_F ! is compared to M*N*EPS. F2 = F CALL DGEMM( 'N', 'N', M, N+1, MIN(M,N+1), -ONE, F1, & LDF, Y, LDY, ONE, F2, LDF ) TMP_FQR = DLANGE( 'F', M, N+1, F2, LDF, WORK ) / & DLANGE( 'F', M, N+1, F, LDF, WORK ) IF ( TMP_FQR > TOL2 ) THEN WRITE(*,*) 'FAILED! Something was wrong with the run.' NFAIL_F_QR = NFAIL_F_QR + 1 END IF END IF !..... DGEDMDQ check point IF ( LSAME(RESIDS, 'R') ) THEN ! Compare the residuals returned by DGEDMDQ with the ! explicitly computed residuals using the matrix A. ! Compute explicitly Y1 = A*Z CALL DGEMM( 'N', 'N', M, KQ, M, ONE, A, M, Z, M, ZERO, Y1, M ) ! ... and then A*Z(:,i) - LAMBDA(i)*Z(:,i), using the real forms ! of the invariant subspaces that correspond to complex conjugate ! pairs of eigencalues. (See the description of Z in DGEDMDQ) i = 1 DO WHILE ( i <= KQ ) IF ( IEIGQ(i) == ZERO ) THEN ! have a real eigenvalue with real eigenvector CALL DAXPY( M, -REIGQ(i), Z(1,i), 1, Y1(1,i), 1 ) ! Y(1:M,i) = Y(1:M,i) - REIG(i)*Z(1:M,i) RES1(i) = DNRM2( M, Y1(1,i), 1) i = i + 1 ELSE ! Have a complex conjugate pair ! REIG(i) +- sqrt(-1)*IMEIG(i). ! Since all computation is done in real ! arithmetic, the formula for the residual ! is recast for real representation of the ! complex conjugate eigenpair. See the ! description of RES. AB(1,1) = REIGQ(i) AB(2,1) = -IEIGQ(i) AB(1,2) = IEIGQ(i) AB(2,2) = REIGQ(i) CALL DGEMM( 'N', 'N', M, 2, 2, -ONE, Z(1,i), & M, AB, 2, ONE, Y1(1,i), M ) ! BLAS CALL ! Y(1:M,i:i+1) = Y(1:M,i:i+1) - Z(1:M,i:i+1) * AB ! INTRINSIC RES1(i) = DLANGE( 'F', M, 2, Y1(1,i), M, & WORK ) ! LAPACK CALL RES1(i+1) = RES1(i) i = i + 2 END IF END DO TMP = ZERO DO i = 1, KQ TMP = MAX( TMP, ABS(RES(i) - RES1(i)) * & SINGVQX(K)/(ANORM*SINGVQX(1)) ) END DO TMP_REZQ = MAX( TMP_REZQ, TMP ) IF ( TMP > TOL2 ) THEN NFAIL_REZQ = NFAIL_REZQ + 1 WRITE(*,*) '................ DGEDMDQ FAILED!', & 'Check the code for implementation errors.' STOP END IF END IF DO i = 1, KQ LAMBDAQ(i,1) = REIGQ(i) LAMBDAQ(i,2) = IEIGQ(i) END DO DEALLOCATE(WORK) DEALLOCATE(IWORK) END IF ! TEST_QRDMD !====================================================================== END DO ! LWMINOPT !write(*,*) 'LWMINOPT loop completed' END DO ! WHTSVD LOOP !write(*,*) 'WHTSVD loop completed' END DO ! NRNK LOOP !write(*,*) 'NRNK loop completed' END DO ! SCALE LOOP !write(*,*) 'SCALE loop completed' END DO ! JOBF LOOP !write(*,*) 'JOBREF loop completed' END DO ! JOBZ LOOP !write(*,*) 'JOBZ loop completed' END DO ! MODE -6:6 !write(*,*) 'MODE loop completed' END DO ! 1 or 2 trajectories !write(*,*) 'trajectories loop completed' DEALLOCATE(A) DEALLOCATE(AC) DEALLOCATE(DA) DEALLOCATE(DL) DEALLOCATE(F) DEALLOCATE(F1) DEALLOCATE(F2) DEALLOCATE(X) DEALLOCATE(X0) DEALLOCATE(SINGVX) DEALLOCATE(SINGVQX) DEALLOCATE(Y) DEALLOCATE(Y0) DEALLOCATE(Y1) DEALLOCATE(Z) DEALLOCATE(Z1) DEALLOCATE(RES) DEALLOCATE(RES1) DEALLOCATE(RESEX) DEALLOCATE(REIG) DEALLOCATE(IEIG) DEALLOCATE(REIGQ) DEALLOCATE(IEIGQ) DEALLOCATE(REIGA) DEALLOCATE(IEIGA) DEALLOCATE(VA) DEALLOCATE(LAMBDA) DEALLOCATE(LAMBDAQ) DEALLOCATE(EIGA) DEALLOCATE(W) DEALLOCATE(AU) DEALLOCATE(S) !............................................................ ! Generate random M-by-M matrix A. Use DLATMR from END DO ! LLOOP WRITE(*,*) '>>>>>>>>>>>>>>>>>>>>>>>>>>' WRITE(*,*) ' Test summary for DGEDMD :' WRITE(*,*) '>>>>>>>>>>>>>>>>>>>>>>>>>>' WRITE(*,*) IF ( NFAIL_Z_XV == 0 ) THEN WRITE(*,*) '>>>> Z - U*V test PASSED.' ELSE WRITE(*,*) 'Z - U*V test FAILED ', NFAIL_Z_XV, ' time(s)' WRITE(*,*) 'Max error ||Z-U*V||_F was ', TMP_ZXW NFAIL_TOTAL = NFAIL_TOTAL + NFAIL_Z_XV END IF IF ( NFAIL_AU == 0 ) THEN WRITE(*,*) '>>>> A*U test PASSED. ' ELSE WRITE(*,*) 'A*U test FAILED ', NFAIL_AU, ' time(s)' WRITE(*,*) 'Max A*U test adjusted error measure was ', TMP_AU WRITE(*,*) 'It should be up to O(M*N) times EPS, EPS = ', EPS NFAIL_TOTAL = NFAIL_TOTAL + NFAIL_AU END IF IF ( NFAIL_REZ == 0 ) THEN WRITE(*,*) '>>>> Rezidual computation test PASSED.' ELSE WRITE(*,*) 'Rezidual computation test FAILED ', NFAIL_REZ, 'time(s)' WRITE(*,*) 'Max residual computing test adjusted error measure was ', TMP_REZ WRITE(*,*) 'It should be up to O(M*N) times EPS, EPS = ', EPS NFAIL_TOTAL = NFAIL_TOTAL + NFAIL_REZ END IF IF ( NFAIL_TOTAL == 0 ) THEN WRITE(*,*) '>>>> DGEDMD :: ALL TESTS PASSED.' ELSE WRITE(*,*) NFAIL_TOTAL, 'FAILURES!' WRITE(*,*) '>>>>>>>>>>>>>> DGEDMD :: TESTS FAILED. CHECK THE IMPLEMENTATION.' END IF IF ( TEST_QRDMD ) THEN WRITE(*,*) WRITE(*,*) '>>>>>>>>>>>>>>>>>>>>>>>>>>' WRITE(*,*) ' Test summary for DGEDMDQ :' WRITE(*,*) '>>>>>>>>>>>>>>>>>>>>>>>>>>' WRITE(*,*) IF ( NFAIL_SVDIFF == 0 ) THEN WRITE(*,*) '>>>> DGEDMD and DGEDMDQ computed singular & &values test PASSED.' ELSE WRITE(*,*) 'DGEDMD and DGEDMDQ discrepancies in & &the singular values unacceptable ', & NFAIL_SVDIFF, ' times. Test FAILED.' WRITE(*,*) 'The maximal discrepancy in the singular values (relative to the norm) was ', SVDIFF WRITE(*,*) 'It should be up to O(M*N) times EPS, EPS = ', EPS NFAILQ_TOTAL = NFAILQ_TOTAL + NFAIL_SVDIFF END IF IF ( NFAIL_F_QR == 0 ) THEN WRITE(*,*) '>>>> F - Q*R test PASSED.' ELSE WRITE(*,*) 'F - Q*R test FAILED ', NFAIL_F_QR, ' time(s)' WRITE(*,*) 'The largest relative residual was ', TMP_FQR WRITE(*,*) 'It should be up to O(M*N) times EPS, EPS = ', EPS NFAILQ_TOTAL = NFAILQ_TOTAL + NFAIL_F_QR END IF IF ( NFAIL_REZQ == 0 ) THEN WRITE(*,*) '>>>> Rezidual computation test PASSED.' ELSE WRITE(*,*) 'Rezidual computation test FAILED ', NFAIL_REZQ, 'time(s)' WRITE(*,*) 'Max residual computing test adjusted error measure was ', TMP_REZQ WRITE(*,*) 'It should be up to O(M*N) times EPS, EPS = ', EPS NFAILQ_TOTAL = NFAILQ_TOTAL + NFAIL_REZQ END IF IF ( NFAILQ_TOTAL == 0 ) THEN WRITE(*,*) '>>>>>>> DGEDMDQ :: ALL TESTS PASSED.' ELSE WRITE(*,*) NFAILQ_TOTAL, 'FAILURES!' WRITE(*,*) '>>>>>>> DGEDMDQ :: TESTS FAILED. CHECK THE IMPLEMENTATION.' END IF END IF WRITE(*,*) WRITE(*,*) 'Test completed.' STOP END