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792 lines
28 KiB
792 lines
28 KiB
! This is a test program for checking the implementations of
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! the implementations of the following subroutines
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!
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! SGEDMD for computation of the
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! Dynamic Mode Decomposition (DMD)
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! SGEDMDQ for computation of a
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! QR factorization based compressed DMD
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!
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! Developed and supported by:
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! ===========================
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! Developed and coded by Zlatko Drmac, Faculty of Science,
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! University of Zagreb; drmac@math.hr
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! In cooperation with
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! AIMdyn Inc., Santa Barbara, CA.
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! ========================================================
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! How to run the code (compiler, link info)
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! ========================================================
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! Compile as FORTRAN 90 (or later) and link with BLAS and
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! LAPACK libraries.
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! NOTE: The code is developed and tested on top of the
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! Intel MKL library (versions 2022.0.3 and 2022.2.0),
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! using the Intel Fortran compiler.
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!
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! For developers of the C++ implementation
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! ========================================================
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! See the LAPACK++ and Template Numerical Toolkit (TNT)
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!
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! Note on a development of the GPU HP implementation
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! ========================================================
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! Work in progress. See CUDA, MAGMA, SLATE.
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! NOTE: The four SVD subroutines used in this code are
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! included as a part of R&D and for the completeness.
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! This was also an opportunity to test those SVD codes.
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! If the scaling option is used all four are essentially
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! equally good. For implementations on HP platforms,
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! one can use whichever SVD is available.
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!... .........................................................
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! NOTE:
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! When using the Intel MKL 2022.0.3 the subroutine xGESVDQ
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! (optionally used in xGEDMD) may cause access violation
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! error for x = S, D, C, Z, but only if called with the
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! work space query. (At least in our Windows 10 MSVS 2019.)
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! The problem can be mitigated by downloading the source
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! code of xGESVDQ from the LAPACK repository and use it
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! localy instead of the one in the MKL. This seems to
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! indicate that the problem is indeed in the MKL.
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! This problem did not appear whith Intel MKL 2022.2.0.
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!
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! NOTE:
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! xGESDD seems to have a problem with workspace. In some
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! cases the length of the optimal workspace is returned
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! smaller than the minimal workspace, as specified in the
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! code. As a precaution, all optimal workspaces are
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! set as MAX(minimal, optimal).
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! Latest implementations of complex xGESDD have different
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! length of the real worksapce. We use max value over
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! two versions.
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!............................................................
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!............................................................
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!
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PROGRAM DMD_TEST
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use iso_fortran_env, only: real32
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IMPLICIT NONE
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integer, parameter :: WP = real32
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!............................................................
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REAL(KIND=WP), PARAMETER :: ONE = 1.0_WP
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REAL(KIND=WP), PARAMETER :: ZERO = 0.0_WP
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!............................................................
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REAL(KIND=WP), ALLOCATABLE, DIMENSION(:,:) :: &
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A, AC, EIGA, LAMBDA, LAMBDAQ, F, F1, F2,&
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Z, Z1, S, AU, W, VA, X, X0, Y, Y0, Y1
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REAL(KIND=WP), ALLOCATABLE, DIMENSION(:) :: &
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DA, DL, DR, REIG, REIGA, REIGQ, IEIG, &
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IEIGA, IEIGQ, RES, RES1, RESEX, SINGVX,&
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SINGVQX, WORK
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INTEGER , ALLOCATABLE, DIMENSION(:) :: IWORK
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REAL(KIND=WP) :: AB(2,2), WDUMMY(2)
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INTEGER :: IDUMMY(2), ISEED(4), RJOBDATA(8)
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REAL(KIND=WP) :: ANORM, COND, CONDL, CONDR, DMAX, EPS, &
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TOL, TOL2, SVDIFF, TMP, TMP_AU, &
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TMP_FQR, TMP_REZ, TMP_REZQ, TMP_ZXW, &
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TMP_EX, XNORM, YNORM
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!............................................................
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INTEGER :: K, KQ, LDF, LDS, LDA, LDAU, LDW, LDX, LDY, &
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LDZ, LIWORK, LWORK, M, N, L, LLOOP, NRNK
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INTEGER :: i, iJOBREF, iJOBZ, iSCALE, INFO, KDIFF, &
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NFAIL, NFAIL_AU, NFAIL_F_QR, NFAIL_REZ, &
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NFAIL_REZQ, NFAIL_SVDIFF, NFAIL_TOTAL, NFAILQ_TOTAL, &
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NFAIL_Z_XV, MODE, MODEL, MODER, WHTSVD
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INTEGER iNRNK, iWHTSVD, K_TRAJ, LWMINOPT
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CHARACTER(LEN=1) GRADE, JOBREF, JOBZ, PIVTNG, RSIGN, &
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SCALE, RESIDS, WANTQ, WANTR
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LOGICAL TEST_QRDMD
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!..... external subroutines (BLAS and LAPACK)
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EXTERNAL SAXPY, SGEEV, SGEMM, SGEMV, SLACPY, SLASCL
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EXTERNAL SLARNV, SLATMR
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!.....external subroutines DMD package, part 1
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! subroutines under test
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EXTERNAL SGEDMD, SGEDMDQ
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!..... external functions (BLAS and LAPACK)
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EXTERNAL SLAMCH, SLANGE, SNRM2
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REAL(KIND=WP) :: SLAMCH, SLANGE, SNRM2
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EXTERNAL LSAME
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LOGICAL LSAME
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INTRINSIC ABS, INT, MIN, MAX
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!............................................................
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! The test is always in pairs : ( SGEDMD and SGEDMDQ )
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! because the test includes comparing the results (in pairs).
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!.....................................................................................
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TEST_QRDMD = .TRUE. ! This code by default performs tests on SGEDMDQ
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! Since the QR factorizations based algorithm is designed for
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! single trajectory data, only single trajectory tests will
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! be performed with xGEDMDQ.
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WANTQ = 'Q'
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WANTR = 'R'
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!.................................................................................
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EPS = SLAMCH( 'P' ) ! machine precision SP
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! Global counters of failures of some particular tests
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NFAIL = 0
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NFAIL_REZ = 0
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NFAIL_REZQ = 0
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NFAIL_Z_XV = 0
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NFAIL_F_QR = 0
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NFAIL_AU = 0
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KDIFF = 0
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NFAIL_SVDIFF = 0
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NFAIL_TOTAL = 0
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NFAILQ_TOTAL = 0
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DO LLOOP = 1, 4
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WRITE(*,*) 'L Loop Index = ', LLOOP
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! Set the dimensions of the problem ...
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WRITE(*,*) 'M = '
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READ(*,*) M
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WRITE(*,*) M
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! ... and the number of snapshots.
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WRITE(*,*) 'N = '
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READ(*,*) N
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WRITE(*,*) N
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! ... Test the dimensions
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IF ( ( MIN(M,N) == 0 ) .OR. ( M < N ) ) THEN
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WRITE(*,*) 'Bad dimensions. Required: M >= N > 0.'
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STOP
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END IF
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!.............
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! The seed inside the LLOOP so that each pass can be reproduced easily.
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ISEED(1) = 4
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ISEED(2) = 3
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ISEED(3) = 2
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ISEED(4) = 1
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LDA = M
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LDF = M
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LDX = MAX(M,N+1)
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LDY = MAX(M,N+1)
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LDW = N
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LDZ = M
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LDAU = MAX(M,N+1)
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LDS = N
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TMP_ZXW = ZERO
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TMP_AU = ZERO
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TMP_REZ = ZERO
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TMP_REZQ = ZERO
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SVDIFF = ZERO
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TMP_EX = ZERO
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!
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! Test the subroutines on real data snapshots. All
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! computation is done in real arithmetic, even when
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! Koopman eigenvalues and modes are real.
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!
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! Allocate memory space
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ALLOCATE( A(LDA,M) )
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ALLOCATE( AC(LDA,M) )
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ALLOCATE( DA(M) )
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ALLOCATE( DL(M) )
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ALLOCATE( F(LDF,N+1) )
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ALLOCATE( F1(LDF,N+1) )
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ALLOCATE( F2(LDF,N+1) )
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ALLOCATE( X(LDX,N) )
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ALLOCATE( X0(LDX,N) )
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ALLOCATE( SINGVX(N) )
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ALLOCATE( SINGVQX(N) )
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ALLOCATE( Y(LDY,N+1) )
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ALLOCATE( Y0(LDY,N+1) )
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ALLOCATE( Y1(M,N+1) )
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ALLOCATE( Z(LDZ,N) )
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ALLOCATE( Z1(LDZ,N) )
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ALLOCATE( RES(N) )
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ALLOCATE( RES1(N) )
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ALLOCATE( RESEX(N) )
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ALLOCATE( REIG(N) )
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ALLOCATE( IEIG(N) )
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ALLOCATE( REIGQ(N) )
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ALLOCATE( IEIGQ(N) )
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ALLOCATE( REIGA(M) )
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ALLOCATE( IEIGA(M) )
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ALLOCATE( VA(LDA,M) )
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ALLOCATE( LAMBDA(N,2) )
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ALLOCATE( LAMBDAQ(N,2) )
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ALLOCATE( EIGA(M,2) )
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ALLOCATE( W(LDW,N) )
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ALLOCATE( AU(LDAU,N) )
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ALLOCATE( S(N,N) )
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TOL = M*EPS
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! This mimics O(M*N)*EPS bound for accumulated roundoff error.
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! The factor 10 is somewhat arbitrary.
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TOL2 = 10*M*N*EPS
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!.............
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DO K_TRAJ = 1, 2
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! Number of intial conditions in the simulation/trajectories (1 or 2)
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COND = 1.0D8
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DMAX = 1.0D2
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RSIGN = 'F'
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GRADE = 'N'
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MODEL = 6
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CONDL = 1.0D2
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MODER = 6
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CONDR = 1.0D2
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PIVTNG = 'N'
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! Loop over all parameter MODE values for ZLATMR (+1,..,+6)
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DO MODE = 1, 6
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ALLOCATE( IWORK(2*M) )
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ALLOCATE(DR(N))
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CALL SLATMR( M, M, 'S', ISEED, 'N', DA, MODE, COND, &
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DMAX, RSIGN, GRADE, DL, MODEL, CONDL, &
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DR, MODER, CONDR, PIVTNG, IWORK, M, M, &
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ZERO, -ONE, 'N', A, LDA, IWORK(M+1), INFO )
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DEALLOCATE(IWORK)
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DEALLOCATE(DR)
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LWORK = 4*M+1
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ALLOCATE(WORK(LWORK))
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AC = A
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CALL SGEEV( 'N','V', M, AC, M, REIGA, IEIGA, VA, M, &
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VA, M, WORK, LWORK, INFO ) ! LAPACK CALL
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DEALLOCATE(WORK)
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TMP = ZERO
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DO i = 1, M
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EIGA(i,1) = REIGA(i)
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EIGA(i,2) = IEIGA(i)
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TMP = MAX( TMP, SQRT(REIGA(i)**2+IEIGA(i)**2))
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END DO
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! Scale A to have the desirable spectral radius.
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CALL SLASCL( 'G', 0, 0, TMP, ONE, M, M, A, M, INFO )
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CALL SLASCL( 'G', 0, 0, TMP, ONE, M, 2, EIGA, M, INFO )
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! Compute the norm of A
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ANORM = SLANGE( 'F', N, N, A, M, WDUMMY )
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IF ( K_TRAJ == 2 ) THEN
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! generate data with two inital conditions
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CALL SLARNV(2, ISEED, M, F1(1,1) )
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F1(1:M,1) = 1.0E-10*F1(1:M,1)
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DO i = 1, N/2
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CALL SGEMV( 'N', M, M, ONE, A, M, F1(1,i), 1, ZERO, &
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F1(1,i+1), 1 )
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END DO
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X0(1:M,1:N/2) = F1(1:M,1:N/2)
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Y0(1:M,1:N/2) = F1(1:M,2:N/2+1)
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CALL SLARNV(2, ISEED, M, F1(1,1) )
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DO i = 1, N-N/2
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CALL SGEMV( 'N', M, M, ONE, A, M, F1(1,i), 1, ZERO, &
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F1(1,i+1), 1 )
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END DO
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X0(1:M,N/2+1:N) = F1(1:M,1:N-N/2)
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Y0(1:M,N/2+1:N) = F1(1:M,2:N-N/2+1)
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ELSE
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! single trajectory
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CALL SLARNV(2, ISEED, M, F(1,1) )
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DO i = 1, N
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CALL SGEMV( 'N', M, M, ONE, A, M, F(1,i), 1, ZERO, &
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F(1,i+1), 1 )
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END DO
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X0(1:M,1:N) = F(1:M,1:N)
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Y0(1:M,1:N) = F(1:M,2:N+1)
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END IF
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XNORM = SLANGE( 'F', M, N, X0, LDX, WDUMMY )
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YNORM = SLANGE( 'F', M, N, Y0, LDX, WDUMMY )
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!............................................................
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DO iJOBZ = 1, 4
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SELECT CASE ( iJOBZ )
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CASE(1)
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JOBZ = 'V' ! Ritz vectors will be computed
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RESIDS = 'R' ! Residuals will be computed
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CASE(2)
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JOBZ = 'V'
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RESIDS = 'N'
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CASE(3)
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JOBZ = 'F' ! Ritz vectors in factored form
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RESIDS = 'N'
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CASE(4)
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JOBZ = 'N'
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RESIDS = 'N'
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END SELECT
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DO iJOBREF = 1, 3
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SELECT CASE ( iJOBREF )
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CASE(1)
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JOBREF = 'R' ! Data for refined Ritz vectors
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CASE(2)
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JOBREF = 'E' ! Exact DMD vectors
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CASE(3)
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JOBREF = 'N'
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END SELECT
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DO iSCALE = 1, 4
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SELECT CASE ( iSCALE )
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CASE(1)
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SCALE = 'S' ! X data normalized
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CASE(2)
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SCALE = 'C' ! X normalized, consist. check
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CASE(3)
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SCALE = 'Y' ! Y data normalized
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CASE(4)
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SCALE = 'N'
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END SELECT
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DO iNRNK = -1, -2, -1
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! Two truncation strategies. The "-2" case for R&D
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! purposes only - it uses possibly low accuracy small
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! singular values, in which case the formulas used in
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! the DMD are highly sensitive.
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NRNK = iNRNK
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DO iWHTSVD = 1, 4
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! Check all four options to compute the POD basis
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! via the SVD.
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WHTSVD = iWHTSVD
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DO LWMINOPT = 1, 2
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! Workspace query for the minimal (1) and for the optimal
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! (2) workspace lengths determined by workspace query.
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X(1:M,1:N) = X0(1:M,1:N)
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Y(1:M,1:N) = Y0(1:M,1:N)
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! SGEDMD: Workspace query and workspace allocation
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CALL SGEDMD( SCALE, JOBZ, RESIDS, JOBREF, WHTSVD, M, &
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N, X, LDX, Y, LDY, NRNK, TOL, K, REIG, IEIG, Z, &
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LDZ, RES, AU, LDAU, W, LDW, S, LDS, WDUMMY, -1, &
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IDUMMY, -1, INFO )
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LIWORK = IDUMMY(1)
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ALLOCATE( IWORK(LIWORK) )
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LWORK = INT(WDUMMY(LWMINOPT))
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ALLOCATE( WORK(LWORK) )
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! SGEDMD test: CALL SGEDMD
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CALL SGEDMD( SCALE, JOBZ, RESIDS, JOBREF, WHTSVD, M, &
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N, X, LDX, Y, LDY, NRNK, TOL, K, REIG, IEIG, Z, &
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LDZ, RES, AU, LDAU, W, LDW, S, LDS, WORK, LWORK,&
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IWORK, LIWORK, INFO )
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SINGVX(1:N) = WORK(1:N)
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!...... SGEDMD check point
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IF ( LSAME(JOBZ,'V') ) THEN
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! Check that Z = X*W, on return from SGEDMD
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! This checks that the returned aigenvectors in Z are
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! the product of the SVD'POD basis returned in X
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! and the eigenvectors of the rayleigh quotient
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! returned in W
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CALL SGEMM( 'N', 'N', M, K, K, ONE, X, LDX, W, LDW, &
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ZERO, Z1, LDZ )
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TMP = ZERO
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DO i = 1, K
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CALL SAXPY( M, -ONE, Z(1,i), 1, Z1(1,i), 1)
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TMP = MAX(TMP, SNRM2( M, Z1(1,i), 1 ) )
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END DO
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TMP_ZXW = MAX(TMP_ZXW, TMP )
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IF ( TMP_ZXW > 10*M*EPS ) THEN
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NFAIL_Z_XV = NFAIL_Z_XV + 1
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END IF
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END IF
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!...... SGEDMD check point
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IF ( LSAME(JOBREF,'R') ) THEN
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! The matrix A*U is returned for computing refined Ritz vectors.
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! Check that A*U is computed correctly using the formula
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! A*U = Y * V * inv(SIGMA). This depends on the
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! accuracy in the computed singular values and vectors of X.
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! See the paper for an error analysis.
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! Note that the left singular vectors of the input matrix X
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! are returned in the array X.
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CALL SGEMM( 'N', 'N', M, K, M, ONE, A, LDA, X, LDX, &
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ZERO, Z1, LDZ )
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TMP = ZERO
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DO i = 1, K
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CALL SAXPY( M, -ONE, AU(1,i), 1, Z1(1,i), 1)
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TMP = MAX( TMP, SNRM2( M, Z1(1,i),1 ) * &
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SINGVX(K)/(ANORM*SINGVX(1)) )
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END DO
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TMP_AU = MAX( TMP_AU, TMP )
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IF ( TMP > TOL2 ) THEN
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NFAIL_AU = NFAIL_AU + 1
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END IF
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ELSEIF ( LSAME(JOBREF,'E') ) THEN
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! The unscaled vectors of the Exact DMD are computed.
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! This option is included for the sake of completeness,
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! for users who prefer the Exact DMD vectors. The
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! returned vectors are in the real form, in the same way
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! as the Ritz vectors. Here we just save the vectors
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! and test them separately using a Matlab script.
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CALL SGEMM( 'N', 'N', M, K, M, ONE, A, LDA, AU, LDAU, ZERO, Y1, M )
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i=1
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DO WHILE ( i <= K )
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IF ( IEIG(i) == ZERO ) THEN
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! have a real eigenvalue with real eigenvector
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CALL SAXPY( M, -REIG(i), AU(1,i), 1, Y1(1,i), 1 )
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RESEX(i) = SNRM2( M, Y1(1,i), 1) / SNRM2(M,AU(1,i),1)
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i = i + 1
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ELSE
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! Have a complex conjugate pair
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! REIG(i) +- sqrt(-1)*IMEIG(i).
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! Since all computation is done in real
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! arithmetic, the formula for the residual
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! is recast for real representation of the
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! 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 SGEMM( 'N', 'N', M, 2, 2, -ONE, AU(1,i), &
|
|
M, AB, 2, ONE, Y1(1,i), M )
|
|
RESEX(i) = SLANGE( 'F', M, 2, Y1(1,i), M, &
|
|
WORK )/ SLANGE( 'F', M, 2, AU(1,i), M, &
|
|
WORK )
|
|
RESEX(i+1) = RESEX(i)
|
|
i = i + 2
|
|
END IF
|
|
END DO
|
|
|
|
END IF
|
|
|
|
!...... SGEDMD check point
|
|
IF ( LSAME(RESIDS, 'R') ) THEN
|
|
! Compare the residuals returned by SGEDMD with the
|
|
! explicitly computed residuals using the matrix A.
|
|
! Compute explicitly Y1 = A*Z
|
|
CALL SGEMM( '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 SGEDMD,)
|
|
i = 1
|
|
DO WHILE ( i <= K )
|
|
IF ( IEIG(i) == ZERO ) THEN
|
|
! have a real eigenvalue with real eigenvector
|
|
CALL SAXPY( M, -REIG(i), Z(1,i), 1, Y1(1,i), 1 )
|
|
RES1(i) = SNRM2( 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 SGEMM( 'N', 'N', M, 2, 2, -ONE, Z(1,i), &
|
|
M, AB, 2, ONE, Y1(1,i), M )
|
|
RES1(i) = SLANGE( '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
|
|
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 SGEDMDQ, if requested.
|
|
!======================================================================
|
|
IF ( TEST_QRDMD .AND. (K_TRAJ == 1) ) THEN
|
|
RJOBDATA(2) = 1
|
|
F1 = F
|
|
|
|
! SGEDMDQ test: Workspace query and workspace allocation
|
|
CALL SGEDMDQ( 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))
|
|
|
|
! SGEDMDQ test: CALL SGEDMDQ
|
|
CALL SGEDMDQ( 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)
|
|
|
|
!..... SGEDMDQ 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
|
|
NFAIL_SVDIFF = NFAIL_SVDIFF + 1
|
|
END IF
|
|
|
|
!..... SGEDMDQ 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 SGEMM( 'N', 'N', M, N+1, MIN(M,N+1), -ONE, F1, &
|
|
LDF, Y, LDY, ONE, F2, LDF )
|
|
TMP_FQR = SLANGE( 'F', M, N+1, F2, LDF, WORK ) / &
|
|
SLANGE( 'F', M, N+1, F, LDF, WORK )
|
|
IF ( TMP_FQR > TOL2 ) THEN
|
|
NFAIL_F_QR = NFAIL_F_QR + 1
|
|
END IF
|
|
END IF
|
|
|
|
!..... SGEDMDQ checkpoint
|
|
IF ( LSAME(RESIDS, 'R') ) THEN
|
|
! Compare the residuals returned by SGEDMDQ with the
|
|
! explicitly computed residuals using the matrix A.
|
|
! Compute explicitly Y1 = A*Z
|
|
CALL SGEMM( '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 SGEDMDQ)
|
|
i = 1
|
|
DO WHILE ( i <= KQ )
|
|
IF ( IEIGQ(i) == ZERO ) THEN
|
|
! have a real eigenvalue with real eigenvector
|
|
CALL SAXPY( 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) = SNRM2( 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 SGEMM( '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) = SLANGE( '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
|
|
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 SGEDMD :'
|
|
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(*,*) '>>>> SGEDMD :: ALL TESTS PASSED.'
|
|
ELSE
|
|
WRITE(*,*) NFAIL_TOTAL, 'FAILURES!'
|
|
WRITE(*,*) '>>>>>>>>>>>>>> SGEDMD :: TESTS FAILED. CHECK THE IMPLEMENTATION.'
|
|
END IF
|
|
|
|
IF ( TEST_QRDMD ) THEN
|
|
WRITE(*,*)
|
|
WRITE(*,*) '>>>>>>>>>>>>>>>>>>>>>>>>>>'
|
|
WRITE(*,*) ' Test summary for SGEDMDQ :'
|
|
WRITE(*,*) '>>>>>>>>>>>>>>>>>>>>>>>>>>'
|
|
WRITE(*,*)
|
|
|
|
IF ( NFAIL_SVDIFF == 0 ) THEN
|
|
WRITE(*,*) '>>>> SGEDMD and SGEDMDQ computed singular &
|
|
&values test PASSED.'
|
|
ELSE
|
|
WRITE(*,*) 'SGEDMD and SGEDMDQ 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(*,*) '>>>>>>> SGEDMDQ :: ALL TESTS PASSED.'
|
|
ELSE
|
|
WRITE(*,*) NFAILQ_TOTAL, 'FAILURES!'
|
|
WRITE(*,*) '>>>>>>> SGEDMDQ :: TESTS FAILED. CHECK THE IMPLEMENTATION.'
|
|
END IF
|
|
|
|
END IF
|
|
|
|
WRITE(*,*)
|
|
WRITE(*,*) 'Test completed.'
|
|
STOP
|
|
END
|
|
|