*> \brief \b SCHKBD * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE SCHKBD( NSIZES, MVAL, NVAL, NTYPES, DOTYPE, NRHS, * ISEED, THRESH, A, LDA, BD, BE, S1, S2, X, LDX, * Y, Z, Q, LDQ, PT, LDPT, U, VT, WORK, LWORK, * IWORK, NOUT, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDPT, LDQ, LDX, LWORK, NOUT, NRHS, * $ NSIZES, NTYPES * REAL THRESH * .. * .. Array Arguments .. * LOGICAL DOTYPE( * ) * INTEGER ISEED( 4 ), IWORK( * ), MVAL( * ), NVAL( * ) * REAL A( LDA, * ), BD( * ), BE( * ), PT( LDPT, * ), * $ Q( LDQ, * ), S1( * ), S2( * ), U( LDPT, * ), * $ VT( LDPT, * ), WORK( * ), X( LDX, * ), * $ Y( LDX, * ), Z( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SCHKBD checks the singular value decomposition (SVD) routines. *> *> SGEBRD reduces a real general m by n matrix A to upper or lower *> bidiagonal form B by an orthogonal transformation: Q' * A * P = B *> (or A = Q * B * P'). The matrix B is upper bidiagonal if m >= n *> and lower bidiagonal if m < n. *> *> SORGBR generates the orthogonal matrices Q and P' from SGEBRD. *> Note that Q and P are not necessarily square. *> *> SBDSQR computes the singular value decomposition of the bidiagonal *> matrix B as B = U S V'. It is called three times to compute *> 1) B = U S1 V', where S1 is the diagonal matrix of singular *> values and the columns of the matrices U and V are the left *> and right singular vectors, respectively, of B. *> 2) Same as 1), but the singular values are stored in S2 and the *> singular vectors are not computed. *> 3) A = (UQ) S (P'V'), the SVD of the original matrix A. *> In addition, SBDSQR has an option to apply the left orthogonal matrix *> U to a matrix X, useful in least squares applications. *> *> SBDSDC computes the singular value decomposition of the bidiagonal *> matrix B as B = U S V' using divide-and-conquer. It is called twice *> to compute *> 1) B = U S1 V', where S1 is the diagonal matrix of singular *> values and the columns of the matrices U and V are the left *> and right singular vectors, respectively, of B. *> 2) Same as 1), but the singular values are stored in S2 and the *> singular vectors are not computed. *> *> For each pair of matrix dimensions (M,N) and each selected matrix *> type, an M by N matrix A and an M by NRHS matrix X are generated. *> The problem dimensions are as follows *> A: M x N *> Q: M x min(M,N) (but M x M if NRHS > 0) *> P: min(M,N) x N *> B: min(M,N) x min(M,N) *> U, V: min(M,N) x min(M,N) *> S1, S2 diagonal, order min(M,N) *> X: M x NRHS *> *> For each generated matrix, 14 tests are performed: *> *> Test SGEBRD and SORGBR *> *> (1) | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P' *> *> (2) | I - Q' Q | / ( M ulp ) *> *> (3) | I - PT PT' | / ( N ulp ) *> *> Test SBDSQR on bidiagonal matrix B *> *> (4) | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V' *> *> (5) | Y - U Z | / ( |Y| max(min(M,N),k) ulp ), where Y = Q' X *> and Z = U' Y. *> (6) | I - U' U | / ( min(M,N) ulp ) *> *> (7) | I - VT VT' | / ( min(M,N) ulp ) *> *> (8) S1 contains min(M,N) nonnegative values in decreasing order. *> (Return 0 if true, 1/ULP if false.) *> *> (9) | S1 - S2 | / ( |S1| ulp ), where S2 is computed without *> computing U and V. *> *> (10) 0 if the true singular values of B are within THRESH of *> those in S1. 2*THRESH if they are not. (Tested using *> SSVDCH) *> *> Test SBDSQR on matrix A *> *> (11) | A - (QU) S (VT PT) | / ( |A| max(M,N) ulp ) *> *> (12) | X - (QU) Z | / ( |X| max(M,k) ulp ) *> *> (13) | I - (QU)'(QU) | / ( M ulp ) *> *> (14) | I - (VT PT) (PT'VT') | / ( N ulp ) *> *> Test SBDSDC on bidiagonal matrix B *> *> (15) | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V' *> *> (16) | I - U' U | / ( min(M,N) ulp ) *> *> (17) | I - VT VT' | / ( min(M,N) ulp ) *> *> (18) S1 contains min(M,N) nonnegative values in decreasing order. *> (Return 0 if true, 1/ULP if false.) *> *> (19) | S1 - S2 | / ( |S1| ulp ), where S2 is computed without *> computing U and V. *> The possible matrix types are *> *> (1) The zero matrix. *> (2) The identity matrix. *> *> (3) A diagonal matrix with evenly spaced entries *> 1, ..., ULP and random signs. *> (ULP = (first number larger than 1) - 1 ) *> (4) A diagonal matrix with geometrically spaced entries *> 1, ..., ULP and random signs. *> (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP *> and random signs. *> *> (6) Same as (3), but multiplied by SQRT( overflow threshold ) *> (7) Same as (3), but multiplied by SQRT( underflow threshold ) *> *> (8) A matrix of the form U D V, where U and V are orthogonal and *> D has evenly spaced entries 1, ..., ULP with random signs *> on the diagonal. *> *> (9) A matrix of the form U D V, where U and V are orthogonal and *> D has geometrically spaced entries 1, ..., ULP with random *> signs on the diagonal. *> *> (10) A matrix of the form U D V, where U and V are orthogonal and *> D has "clustered" entries 1, ULP,..., ULP with random *> signs on the diagonal. *> *> (11) Same as (8), but multiplied by SQRT( overflow threshold ) *> (12) Same as (8), but multiplied by SQRT( underflow threshold ) *> *> (13) Rectangular matrix with random entries chosen from (-1,1). *> (14) Same as (13), but multiplied by SQRT( overflow threshold ) *> (15) Same as (13), but multiplied by SQRT( underflow threshold ) *> *> Special case: *> (16) A bidiagonal matrix with random entries chosen from a *> logarithmic distribution on [ulp^2,ulp^(-2)] (I.e., each *> entry is e^x, where x is chosen uniformly on *> [ 2 log(ulp), -2 log(ulp) ] .) For *this* type: *> (a) SGEBRD is not called to reduce it to bidiagonal form. *> (b) the bidiagonal is min(M,N) x min(M,N); if M matrix will be lower bidiagonal, otherwise upper. *> (c) only tests 5--8 and 14 are performed. *> *> A subset of the full set of matrix types may be selected through *> the logical array DOTYPE. *> \endverbatim * * Arguments: * ========== * *> \param[in] NSIZES *> \verbatim *> NSIZES is INTEGER *> The number of values of M and N contained in the vectors *> MVAL and NVAL. The matrix sizes are used in pairs (M,N). *> \endverbatim *> *> \param[in] MVAL *> \verbatim *> MVAL is INTEGER array, dimension (NM) *> The values of the matrix row dimension M. *> \endverbatim *> *> \param[in] NVAL *> \verbatim *> NVAL is INTEGER array, dimension (NM) *> The values of the matrix column dimension N. *> \endverbatim *> *> \param[in] NTYPES *> \verbatim *> NTYPES is INTEGER *> The number of elements in DOTYPE. If it is zero, SCHKBD *> does nothing. It must be at least zero. If it is MAXTYP+1 *> and NSIZES is 1, then an additional type, MAXTYP+1 is *> defined, which is to use whatever matrices are in A and B. *> This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and *> DOTYPE(MAXTYP+1) is .TRUE. . *> \endverbatim *> *> \param[in] DOTYPE *> \verbatim *> DOTYPE is LOGICAL array, dimension (NTYPES) *> If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix *> of type j will be generated. If NTYPES is smaller than the *> maximum number of types defined (PARAMETER MAXTYP), then *> types NTYPES+1 through MAXTYP will not be generated. If *> NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through *> DOTYPE(NTYPES) will be ignored. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of columns in the "right-hand side" matrices X, Y, *> and Z, used in testing SBDSQR. If NRHS = 0, then the *> operations on the right-hand side will not be tested. *> NRHS must be at least 0. *> \endverbatim *> *> \param[in,out] ISEED *> \verbatim *> ISEED is INTEGER array, dimension (4) *> On entry ISEED specifies the seed of the random number *> generator. The array elements should be between 0 and 4095; *> if not they will be reduced mod 4096. Also, ISEED(4) must *> be odd. The values of ISEED are changed on exit, and can be *> used in the next call to SCHKBD to continue the same random *> number sequence. *> \endverbatim *> *> \param[in] THRESH *> \verbatim *> THRESH is REAL *> The threshold value for the test ratios. A result is *> included in the output file if RESULT >= THRESH. To have *> every test ratio printed, use THRESH = 0. Note that the *> expected value of the test ratios is O(1), so THRESH should *> be a reasonably small multiple of 1, e.g., 10 or 100. *> \endverbatim *> *> \param[out] A *> \verbatim *> A is REAL array, dimension (LDA,NMAX) *> where NMAX is the maximum value of N in NVAL. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,MMAX), *> where MMAX is the maximum value of M in MVAL. *> \endverbatim *> *> \param[out] BD *> \verbatim *> BD is REAL array, dimension *> (max(min(MVAL(j),NVAL(j)))) *> \endverbatim *> *> \param[out] BE *> \verbatim *> BE is REAL array, dimension *> (max(min(MVAL(j),NVAL(j)))) *> \endverbatim *> *> \param[out] S1 *> \verbatim *> S1 is REAL array, dimension *> (max(min(MVAL(j),NVAL(j)))) *> \endverbatim *> *> \param[out] S2 *> \verbatim *> S2 is REAL array, dimension *> (max(min(MVAL(j),NVAL(j)))) *> \endverbatim *> *> \param[out] X *> \verbatim *> X is REAL array, dimension (LDX,NRHS) *> \endverbatim *> *> \param[in] LDX *> \verbatim *> LDX is INTEGER *> The leading dimension of the arrays X, Y, and Z. *> LDX >= max(1,MMAX) *> \endverbatim *> *> \param[out] Y *> \verbatim *> Y is REAL array, dimension (LDX,NRHS) *> \endverbatim *> *> \param[out] Z *> \verbatim *> Z is REAL array, dimension (LDX,NRHS) *> \endverbatim *> *> \param[out] Q *> \verbatim *> Q is REAL array, dimension (LDQ,MMAX) *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. LDQ >= max(1,MMAX). *> \endverbatim *> *> \param[out] PT *> \verbatim *> PT is REAL array, dimension (LDPT,NMAX) *> \endverbatim *> *> \param[in] LDPT *> \verbatim *> LDPT is INTEGER *> The leading dimension of the arrays PT, U, and V. *> LDPT >= max(1, max(min(MVAL(j),NVAL(j)))). *> \endverbatim *> *> \param[out] U *> \verbatim *> U is REAL array, dimension *> (LDPT,max(min(MVAL(j),NVAL(j)))) *> \endverbatim *> *> \param[out] VT *> \verbatim *> VT is REAL array, dimension *> (LDPT,max(min(MVAL(j),NVAL(j)))) *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (LWORK) *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The number of entries in WORK. This must be at least *> 3(M+N) and M(M + max(M,N,k) + 1) + N*min(M,N) for all *> pairs (M,N)=(MM(j),NN(j)) *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension at least 8*min(M,N) *> \endverbatim *> *> \param[in] NOUT *> \verbatim *> NOUT is INTEGER *> The FORTRAN unit number for printing out error messages *> (e.g., if a routine returns IINFO not equal to 0.) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> If 0, then everything ran OK. *> -1: NSIZES < 0 *> -2: Some MM(j) < 0 *> -3: Some NN(j) < 0 *> -4: NTYPES < 0 *> -6: NRHS < 0 *> -8: THRESH < 0 *> -11: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ). *> -17: LDB < 1 or LDB < MMAX. *> -21: LDQ < 1 or LDQ < MMAX. *> -23: LDPT< 1 or LDPT< MNMAX. *> -27: LWORK too small. *> If SLATMR, SLATMS, SGEBRD, SORGBR, or SBDSQR, *> returns an error code, the *> absolute value of it is returned. *> *>----------------------------------------------------------------------- *> *> Some Local Variables and Parameters: *> ---- ----- --------- --- ---------- *> *> ZERO, ONE Real 0 and 1. *> MAXTYP The number of types defined. *> NTEST The number of tests performed, or which can *> be performed so far, for the current matrix. *> MMAX Largest value in NN. *> NMAX Largest value in NN. *> MNMIN min(MM(j), NN(j)) (the dimension of the bidiagonal *> matrix.) *> MNMAX The maximum value of MNMIN for j=1,...,NSIZES. *> NFAIL The number of tests which have exceeded THRESH *> COND, IMODE Values to be passed to the matrix generators. *> ANORM Norm of A; passed to matrix generators. *> *> OVFL, UNFL Overflow and underflow thresholds. *> RTOVFL, RTUNFL Square roots of the previous 2 values. *> ULP, ULPINV Finest relative precision and its inverse. *> *> The following four arrays decode JTYPE: *> KTYPE(j) The general type (1-10) for type "j". *> KMODE(j) The MODE value to be passed to the matrix *> generator for type "j". *> KMAGN(j) The order of magnitude ( O(1), *> O(overflow^(1/2) ), O(underflow^(1/2) ) *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date November 2011 * *> \ingroup single_eig * * ===================================================================== SUBROUTINE SCHKBD( NSIZES, MVAL, NVAL, NTYPES, DOTYPE, NRHS, $ ISEED, THRESH, A, LDA, BD, BE, S1, S2, X, LDX, $ Y, Z, Q, LDQ, PT, LDPT, U, VT, WORK, LWORK, $ IWORK, NOUT, INFO ) * * -- LAPACK test routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2011 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDPT, LDQ, LDX, LWORK, NOUT, NRHS, $ NSIZES, NTYPES REAL THRESH * .. * .. Array Arguments .. LOGICAL DOTYPE( * ) INTEGER ISEED( 4 ), IWORK( * ), MVAL( * ), NVAL( * ) REAL A( LDA, * ), BD( * ), BE( * ), PT( LDPT, * ), $ Q( LDQ, * ), S1( * ), S2( * ), U( LDPT, * ), $ VT( LDPT, * ), WORK( * ), X( LDX, * ), $ Y( LDX, * ), Z( LDX, * ) * .. * * ====================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO, HALF PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0, $ HALF = 0.5E0 ) INTEGER MAXTYP PARAMETER ( MAXTYP = 16 ) * .. * .. Local Scalars .. LOGICAL BADMM, BADNN, BIDIAG CHARACTER UPLO CHARACTER*3 PATH INTEGER I, IINFO, IMODE, ITYPE, J, JCOL, JSIZE, JTYPE, $ LOG2UI, M, MINWRK, MMAX, MNMAX, MNMIN, MQ, $ MTYPES, N, NFAIL, NMAX, NTEST REAL AMNINV, ANORM, COND, OVFL, RTOVFL, RTUNFL, $ TEMP1, TEMP2, ULP, ULPINV, UNFL * .. * .. Local Arrays .. INTEGER IDUM( 1 ), IOLDSD( 4 ), KMAGN( MAXTYP ), $ KMODE( MAXTYP ), KTYPE( MAXTYP ) REAL DUM( 1 ), DUMMA( 1 ), RESULT( 19 ) * .. * .. External Functions .. REAL SLAMCH, SLARND EXTERNAL SLAMCH, SLARND * .. * .. External Subroutines .. EXTERNAL ALASUM, SBDSDC, SBDSQR, SBDT01, SBDT02, SBDT03, $ SCOPY, SGEBRD, SGEMM, SLABAD, SLACPY, SLAHD2, $ SLASET, SLATMR, SLATMS, SORGBR, SORT01, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, EXP, INT, LOG, MAX, MIN, SQRT * .. * .. Scalars in Common .. LOGICAL LERR, OK CHARACTER*32 SRNAMT INTEGER INFOT, NUNIT * .. * .. Common blocks .. COMMON / INFOC / INFOT, NUNIT, OK, LERR COMMON / SRNAMC / SRNAMT * .. * .. Data statements .. DATA KTYPE / 1, 2, 5*4, 5*6, 3*9, 10 / DATA KMAGN / 2*1, 3*1, 2, 3, 3*1, 2, 3, 1, 2, 3, 0 / DATA KMODE / 2*0, 4, 3, 1, 4, 4, 4, 3, 1, 4, 4, 0, $ 0, 0, 0 / * .. * .. Executable Statements .. * * Check for errors * INFO = 0 * BADMM = .FALSE. BADNN = .FALSE. MMAX = 1 NMAX = 1 MNMAX = 1 MINWRK = 1 DO 10 J = 1, NSIZES MMAX = MAX( MMAX, MVAL( J ) ) IF( MVAL( J ).LT.0 ) $ BADMM = .TRUE. NMAX = MAX( NMAX, NVAL( J ) ) IF( NVAL( J ).LT.0 ) $ BADNN = .TRUE. MNMAX = MAX( MNMAX, MIN( MVAL( J ), NVAL( J ) ) ) MINWRK = MAX( MINWRK, 3*( MVAL( J )+NVAL( J ) ), $ MVAL( J )*( MVAL( J )+MAX( MVAL( J ), NVAL( J ), $ NRHS )+1 )+NVAL( J )*MIN( NVAL( J ), MVAL( J ) ) ) 10 CONTINUE * * Check for errors * IF( NSIZES.LT.0 ) THEN INFO = -1 ELSE IF( BADMM ) THEN INFO = -2 ELSE IF( BADNN ) THEN INFO = -3 ELSE IF( NTYPES.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -6 ELSE IF( LDA.LT.MMAX ) THEN INFO = -11 ELSE IF( LDX.LT.MMAX ) THEN INFO = -17 ELSE IF( LDQ.LT.MMAX ) THEN INFO = -21 ELSE IF( LDPT.LT.MNMAX ) THEN INFO = -23 ELSE IF( MINWRK.GT.LWORK ) THEN INFO = -27 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SCHKBD', -INFO ) RETURN END IF * * Initialize constants * PATH( 1: 1 ) = 'Single precision' PATH( 2: 3 ) = 'BD' NFAIL = 0 NTEST = 0 UNFL = SLAMCH( 'Safe minimum' ) OVFL = SLAMCH( 'Overflow' ) CALL SLABAD( UNFL, OVFL ) ULP = SLAMCH( 'Precision' ) ULPINV = ONE / ULP LOG2UI = INT( LOG( ULPINV ) / LOG( TWO ) ) RTUNFL = SQRT( UNFL ) RTOVFL = SQRT( OVFL ) INFOT = 0 * * Loop over sizes, types * DO 200 JSIZE = 1, NSIZES M = MVAL( JSIZE ) N = NVAL( JSIZE ) MNMIN = MIN( M, N ) AMNINV = ONE / MAX( M, N, 1 ) * IF( NSIZES.NE.1 ) THEN MTYPES = MIN( MAXTYP, NTYPES ) ELSE MTYPES = MIN( MAXTYP+1, NTYPES ) END IF * DO 190 JTYPE = 1, MTYPES IF( .NOT.DOTYPE( JTYPE ) ) $ GO TO 190 * DO 20 J = 1, 4 IOLDSD( J ) = ISEED( J ) 20 CONTINUE * DO 30 J = 1, 14 RESULT( J ) = -ONE 30 CONTINUE * UPLO = ' ' * * Compute "A" * * Control parameters: * * KMAGN KMODE KTYPE * =1 O(1) clustered 1 zero * =2 large clustered 2 identity * =3 small exponential (none) * =4 arithmetic diagonal, (w/ eigenvalues) * =5 random symmetric, w/ eigenvalues * =6 nonsymmetric, w/ singular values * =7 random diagonal * =8 random symmetric * =9 random nonsymmetric * =10 random bidiagonal (log. distrib.) * IF( MTYPES.GT.MAXTYP ) $ GO TO 100 * ITYPE = KTYPE( JTYPE ) IMODE = KMODE( JTYPE ) * * Compute norm * GO TO ( 40, 50, 60 )KMAGN( JTYPE ) * 40 CONTINUE ANORM = ONE GO TO 70 * 50 CONTINUE ANORM = ( RTOVFL*ULP )*AMNINV GO TO 70 * 60 CONTINUE ANORM = RTUNFL*MAX( M, N )*ULPINV GO TO 70 * 70 CONTINUE * CALL SLASET( 'Full', LDA, N, ZERO, ZERO, A, LDA ) IINFO = 0 COND = ULPINV * BIDIAG = .FALSE. IF( ITYPE.EQ.1 ) THEN * * Zero matrix * IINFO = 0 * ELSE IF( ITYPE.EQ.2 ) THEN * * Identity * DO 80 JCOL = 1, MNMIN A( JCOL, JCOL ) = ANORM 80 CONTINUE * ELSE IF( ITYPE.EQ.4 ) THEN * * Diagonal Matrix, [Eigen]values Specified * CALL SLATMS( MNMIN, MNMIN, 'S', ISEED, 'N', WORK, IMODE, $ COND, ANORM, 0, 0, 'N', A, LDA, $ WORK( MNMIN+1 ), IINFO ) * ELSE IF( ITYPE.EQ.5 ) THEN * * Symmetric, eigenvalues specified * CALL SLATMS( MNMIN, MNMIN, 'S', ISEED, 'S', WORK, IMODE, $ COND, ANORM, M, N, 'N', A, LDA, $ WORK( MNMIN+1 ), IINFO ) * ELSE IF( ITYPE.EQ.6 ) THEN * * Nonsymmetric, singular values specified * CALL SLATMS( M, N, 'S', ISEED, 'N', WORK, IMODE, COND, $ ANORM, M, N, 'N', A, LDA, WORK( MNMIN+1 ), $ IINFO ) * ELSE IF( ITYPE.EQ.7 ) THEN * * Diagonal, random entries * CALL SLATMR( MNMIN, MNMIN, 'S', ISEED, 'N', WORK, 6, ONE, $ ONE, 'T', 'N', WORK( MNMIN+1 ), 1, ONE, $ WORK( 2*MNMIN+1 ), 1, ONE, 'N', IWORK, 0, 0, $ ZERO, ANORM, 'NO', A, LDA, IWORK, IINFO ) * ELSE IF( ITYPE.EQ.8 ) THEN * * Symmetric, random entries * CALL SLATMR( MNMIN, MNMIN, 'S', ISEED, 'S', WORK, 6, ONE, $ ONE, 'T', 'N', WORK( MNMIN+1 ), 1, ONE, $ WORK( M+MNMIN+1 ), 1, ONE, 'N', IWORK, M, N, $ ZERO, ANORM, 'NO', A, LDA, IWORK, IINFO ) * ELSE IF( ITYPE.EQ.9 ) THEN * * Nonsymmetric, random entries * CALL SLATMR( M, N, 'S', ISEED, 'N', WORK, 6, ONE, ONE, $ 'T', 'N', WORK( MNMIN+1 ), 1, ONE, $ WORK( M+MNMIN+1 ), 1, ONE, 'N', IWORK, M, N, $ ZERO, ANORM, 'NO', A, LDA, IWORK, IINFO ) * ELSE IF( ITYPE.EQ.10 ) THEN * * Bidiagonal, random entries * TEMP1 = -TWO*LOG( ULP ) DO 90 J = 1, MNMIN BD( J ) = EXP( TEMP1*SLARND( 2, ISEED ) ) IF( J.LT.MNMIN ) $ BE( J ) = EXP( TEMP1*SLARND( 2, ISEED ) ) 90 CONTINUE * IINFO = 0 BIDIAG = .TRUE. IF( M.GE.N ) THEN UPLO = 'U' ELSE UPLO = 'L' END IF ELSE IINFO = 1 END IF * IF( IINFO.EQ.0 ) THEN * * Generate Right-Hand Side * IF( BIDIAG ) THEN CALL SLATMR( MNMIN, NRHS, 'S', ISEED, 'N', WORK, 6, $ ONE, ONE, 'T', 'N', WORK( MNMIN+1 ), 1, $ ONE, WORK( 2*MNMIN+1 ), 1, ONE, 'N', $ IWORK, MNMIN, NRHS, ZERO, ONE, 'NO', Y, $ LDX, IWORK, IINFO ) ELSE CALL SLATMR( M, NRHS, 'S', ISEED, 'N', WORK, 6, ONE, $ ONE, 'T', 'N', WORK( M+1 ), 1, ONE, $ WORK( 2*M+1 ), 1, ONE, 'N', IWORK, M, $ NRHS, ZERO, ONE, 'NO', X, LDX, IWORK, $ IINFO ) END IF END IF * * Error Exit * IF( IINFO.NE.0 ) THEN WRITE( NOUT, FMT = 9998 )'Generator', IINFO, M, N, $ JTYPE, IOLDSD INFO = ABS( IINFO ) RETURN END IF * 100 CONTINUE * * Call SGEBRD and SORGBR to compute B, Q, and P, do tests. * IF( .NOT.BIDIAG ) THEN * * Compute transformations to reduce A to bidiagonal form: * B := Q' * A * P. * CALL SLACPY( ' ', M, N, A, LDA, Q, LDQ ) CALL SGEBRD( M, N, Q, LDQ, BD, BE, WORK, WORK( MNMIN+1 ), $ WORK( 2*MNMIN+1 ), LWORK-2*MNMIN, IINFO ) * * Check error code from SGEBRD. * IF( IINFO.NE.0 ) THEN WRITE( NOUT, FMT = 9998 )'SGEBRD', IINFO, M, N, $ JTYPE, IOLDSD INFO = ABS( IINFO ) RETURN END IF * CALL SLACPY( ' ', M, N, Q, LDQ, PT, LDPT ) IF( M.GE.N ) THEN UPLO = 'U' ELSE UPLO = 'L' END IF * * Generate Q * MQ = M IF( NRHS.LE.0 ) $ MQ = MNMIN CALL SORGBR( 'Q', M, MQ, N, Q, LDQ, WORK, $ WORK( 2*MNMIN+1 ), LWORK-2*MNMIN, IINFO ) * * Check error code from SORGBR. * IF( IINFO.NE.0 ) THEN WRITE( NOUT, FMT = 9998 )'SORGBR(Q)', IINFO, M, N, $ JTYPE, IOLDSD INFO = ABS( IINFO ) RETURN END IF * * Generate P' * CALL SORGBR( 'P', MNMIN, N, M, PT, LDPT, WORK( MNMIN+1 ), $ WORK( 2*MNMIN+1 ), LWORK-2*MNMIN, IINFO ) * * Check error code from SORGBR. * IF( IINFO.NE.0 ) THEN WRITE( NOUT, FMT = 9998 )'SORGBR(P)', IINFO, M, N, $ JTYPE, IOLDSD INFO = ABS( IINFO ) RETURN END IF * * Apply Q' to an M by NRHS matrix X: Y := Q' * X. * CALL SGEMM( 'Transpose', 'No transpose', M, NRHS, M, ONE, $ Q, LDQ, X, LDX, ZERO, Y, LDX ) * * Test 1: Check the decomposition A := Q * B * PT * 2: Check the orthogonality of Q * 3: Check the orthogonality of PT * CALL SBDT01( M, N, 1, A, LDA, Q, LDQ, BD, BE, PT, LDPT, $ WORK, RESULT( 1 ) ) CALL SORT01( 'Columns', M, MQ, Q, LDQ, WORK, LWORK, $ RESULT( 2 ) ) CALL SORT01( 'Rows', MNMIN, N, PT, LDPT, WORK, LWORK, $ RESULT( 3 ) ) END IF * * Use SBDSQR to form the SVD of the bidiagonal matrix B: * B := U * S1 * VT, and compute Z = U' * Y. * CALL SCOPY( MNMIN, BD, 1, S1, 1 ) IF( MNMIN.GT.0 ) $ CALL SCOPY( MNMIN-1, BE, 1, WORK, 1 ) CALL SLACPY( ' ', M, NRHS, Y, LDX, Z, LDX ) CALL SLASET( 'Full', MNMIN, MNMIN, ZERO, ONE, U, LDPT ) CALL SLASET( 'Full', MNMIN, MNMIN, ZERO, ONE, VT, LDPT ) * CALL SBDSQR( UPLO, MNMIN, MNMIN, MNMIN, NRHS, S1, WORK, VT, $ LDPT, U, LDPT, Z, LDX, WORK( MNMIN+1 ), IINFO ) * * Check error code from SBDSQR. * IF( IINFO.NE.0 ) THEN WRITE( NOUT, FMT = 9998 )'SBDSQR(vects)', IINFO, M, N, $ JTYPE, IOLDSD INFO = ABS( IINFO ) IF( IINFO.LT.0 ) THEN RETURN ELSE RESULT( 4 ) = ULPINV GO TO 170 END IF END IF * * Use SBDSQR to compute only the singular values of the * bidiagonal matrix B; U, VT, and Z should not be modified. * CALL SCOPY( MNMIN, BD, 1, S2, 1 ) IF( MNMIN.GT.0 ) $ CALL SCOPY( MNMIN-1, BE, 1, WORK, 1 ) * CALL SBDSQR( UPLO, MNMIN, 0, 0, 0, S2, WORK, VT, LDPT, U, $ LDPT, Z, LDX, WORK( MNMIN+1 ), IINFO ) * * Check error code from SBDSQR. * IF( IINFO.NE.0 ) THEN WRITE( NOUT, FMT = 9998 )'SBDSQR(values)', IINFO, M, N, $ JTYPE, IOLDSD INFO = ABS( IINFO ) IF( IINFO.LT.0 ) THEN RETURN ELSE RESULT( 9 ) = ULPINV GO TO 170 END IF END IF * * Test 4: Check the decomposition B := U * S1 * VT * 5: Check the computation Z := U' * Y * 6: Check the orthogonality of U * 7: Check the orthogonality of VT * CALL SBDT03( UPLO, MNMIN, 1, BD, BE, U, LDPT, S1, VT, LDPT, $ WORK, RESULT( 4 ) ) CALL SBDT02( MNMIN, NRHS, Y, LDX, Z, LDX, U, LDPT, WORK, $ RESULT( 5 ) ) CALL SORT01( 'Columns', MNMIN, MNMIN, U, LDPT, WORK, LWORK, $ RESULT( 6 ) ) CALL SORT01( 'Rows', MNMIN, MNMIN, VT, LDPT, WORK, LWORK, $ RESULT( 7 ) ) * * Test 8: Check that the singular values are sorted in * non-increasing order and are non-negative * RESULT( 8 ) = ZERO DO 110 I = 1, MNMIN - 1 IF( S1( I ).LT.S1( I+1 ) ) $ RESULT( 8 ) = ULPINV IF( S1( I ).LT.ZERO ) $ RESULT( 8 ) = ULPINV 110 CONTINUE IF( MNMIN.GE.1 ) THEN IF( S1( MNMIN ).LT.ZERO ) $ RESULT( 8 ) = ULPINV END IF * * Test 9: Compare SBDSQR with and without singular vectors * TEMP2 = ZERO * DO 120 J = 1, MNMIN TEMP1 = ABS( S1( J )-S2( J ) ) / $ MAX( SQRT( UNFL )*MAX( S1( 1 ), ONE ), $ ULP*MAX( ABS( S1( J ) ), ABS( S2( J ) ) ) ) TEMP2 = MAX( TEMP1, TEMP2 ) 120 CONTINUE * RESULT( 9 ) = TEMP2 * * Test 10: Sturm sequence test of singular values * Go up by factors of two until it succeeds * TEMP1 = THRESH*( HALF-ULP ) * DO 130 J = 0, LOG2UI * CALL SSVDCH( MNMIN, BD, BE, S1, TEMP1, IINFO ) IF( IINFO.EQ.0 ) $ GO TO 140 TEMP1 = TEMP1*TWO 130 CONTINUE * 140 CONTINUE RESULT( 10 ) = TEMP1 * * Use SBDSQR to form the decomposition A := (QU) S (VT PT) * from the bidiagonal form A := Q B PT. * IF( .NOT.BIDIAG ) THEN CALL SCOPY( MNMIN, BD, 1, S2, 1 ) IF( MNMIN.GT.0 ) $ CALL SCOPY( MNMIN-1, BE, 1, WORK, 1 ) * CALL SBDSQR( UPLO, MNMIN, N, M, NRHS, S2, WORK, PT, LDPT, $ Q, LDQ, Y, LDX, WORK( MNMIN+1 ), IINFO ) * * Test 11: Check the decomposition A := Q*U * S2 * VT*PT * 12: Check the computation Z := U' * Q' * X * 13: Check the orthogonality of Q*U * 14: Check the orthogonality of VT*PT * CALL SBDT01( M, N, 0, A, LDA, Q, LDQ, S2, DUMMA, PT, $ LDPT, WORK, RESULT( 11 ) ) CALL SBDT02( M, NRHS, X, LDX, Y, LDX, Q, LDQ, WORK, $ RESULT( 12 ) ) CALL SORT01( 'Columns', M, MQ, Q, LDQ, WORK, LWORK, $ RESULT( 13 ) ) CALL SORT01( 'Rows', MNMIN, N, PT, LDPT, WORK, LWORK, $ RESULT( 14 ) ) END IF * * Use SBDSDC to form the SVD of the bidiagonal matrix B: * B := U * S1 * VT * CALL SCOPY( MNMIN, BD, 1, S1, 1 ) IF( MNMIN.GT.0 ) $ CALL SCOPY( MNMIN-1, BE, 1, WORK, 1 ) CALL SLASET( 'Full', MNMIN, MNMIN, ZERO, ONE, U, LDPT ) CALL SLASET( 'Full', MNMIN, MNMIN, ZERO, ONE, VT, LDPT ) * CALL SBDSDC( UPLO, 'I', MNMIN, S1, WORK, U, LDPT, VT, LDPT, $ DUM, IDUM, WORK( MNMIN+1 ), IWORK, IINFO ) * * Check error code from SBDSDC. * IF( IINFO.NE.0 ) THEN WRITE( NOUT, FMT = 9998 )'SBDSDC(vects)', IINFO, M, N, $ JTYPE, IOLDSD INFO = ABS( IINFO ) IF( IINFO.LT.0 ) THEN RETURN ELSE RESULT( 15 ) = ULPINV GO TO 170 END IF END IF * * Use SBDSDC to compute only the singular values of the * bidiagonal matrix B; U and VT should not be modified. * CALL SCOPY( MNMIN, BD, 1, S2, 1 ) IF( MNMIN.GT.0 ) $ CALL SCOPY( MNMIN-1, BE, 1, WORK, 1 ) * CALL SBDSDC( UPLO, 'N', MNMIN, S2, WORK, DUM, 1, DUM, 1, $ DUM, IDUM, WORK( MNMIN+1 ), IWORK, IINFO ) * * Check error code from SBDSDC. * IF( IINFO.NE.0 ) THEN WRITE( NOUT, FMT = 9998 )'SBDSDC(values)', IINFO, M, N, $ JTYPE, IOLDSD INFO = ABS( IINFO ) IF( IINFO.LT.0 ) THEN RETURN ELSE RESULT( 18 ) = ULPINV GO TO 170 END IF END IF * * Test 15: Check the decomposition B := U * S1 * VT * 16: Check the orthogonality of U * 17: Check the orthogonality of VT * CALL SBDT03( UPLO, MNMIN, 1, BD, BE, U, LDPT, S1, VT, LDPT, $ WORK, RESULT( 15 ) ) CALL SORT01( 'Columns', MNMIN, MNMIN, U, LDPT, WORK, LWORK, $ RESULT( 16 ) ) CALL SORT01( 'Rows', MNMIN, MNMIN, VT, LDPT, WORK, LWORK, $ RESULT( 17 ) ) * * Test 18: Check that the singular values are sorted in * non-increasing order and are non-negative * RESULT( 18 ) = ZERO DO 150 I = 1, MNMIN - 1 IF( S1( I ).LT.S1( I+1 ) ) $ RESULT( 18 ) = ULPINV IF( S1( I ).LT.ZERO ) $ RESULT( 18 ) = ULPINV 150 CONTINUE IF( MNMIN.GE.1 ) THEN IF( S1( MNMIN ).LT.ZERO ) $ RESULT( 18 ) = ULPINV END IF * * Test 19: Compare SBDSQR with and without singular vectors * TEMP2 = ZERO * DO 160 J = 1, MNMIN TEMP1 = ABS( S1( J )-S2( J ) ) / $ MAX( SQRT( UNFL )*MAX( S1( 1 ), ONE ), $ ULP*MAX( ABS( S1( 1 ) ), ABS( S2( 1 ) ) ) ) TEMP2 = MAX( TEMP1, TEMP2 ) 160 CONTINUE * RESULT( 19 ) = TEMP2 * * End of Loop -- Check for RESULT(j) > THRESH * 170 CONTINUE DO 180 J = 1, 19 IF( RESULT( J ).GE.THRESH ) THEN IF( NFAIL.EQ.0 ) $ CALL SLAHD2( NOUT, PATH ) WRITE( NOUT, FMT = 9999 )M, N, JTYPE, IOLDSD, J, $ RESULT( J ) NFAIL = NFAIL + 1 END IF 180 CONTINUE IF( .NOT.BIDIAG ) THEN NTEST = NTEST + 19 ELSE NTEST = NTEST + 5 END IF * 190 CONTINUE 200 CONTINUE * * Summary * CALL ALASUM( PATH, NOUT, NFAIL, NTEST, 0 ) * RETURN * * End of SCHKBD * 9999 FORMAT( ' M=', I5, ', N=', I5, ', type ', I2, ', seed=', $ 4( I4, ',' ), ' test(', I2, ')=', G11.4 ) 9998 FORMAT( ' SCHKBD: ', A, ' returned INFO=', I6, '.', / 9X, 'M=', $ I6, ', N=', I6, ', JTYPE=', I6, ', ISEED=(', 3( I5, ',' ), $ I5, ')' ) * END