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SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
$ M, N, A, LDA, SVA, U, LDU, V, LDV, $ WORK, LWORK, IWORK, INFO ) * * -- LAPACK routine (version 3.3.1) -- * * -- Contributed by Zlatko Drmac of the University of Zagreb and -- * -- Kresimir Veselic of the Fernuniversitaet Hagen -- * -- April 2011 -- * * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * This routine is also part of SIGMA (version 1.23, October 23. 2008.) * SIGMA is a library of algorithms for highly accurate algorithms for * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0. * * .. Scalar Arguments .. IMPLICIT NONE INTEGER INFO, LDA, LDU, LDV, LWORK, M, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ), $ WORK( LWORK ) INTEGER IWORK( * ) CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV * .. * * Purpose * ======= * * DGEJSV computes the singular value decomposition (SVD) of a real M-by-N * matrix [A], where M >= N. The SVD of [A] is written as * * [A] = [U] * [SIGMA] * [V]^t, * * where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N * diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and * [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are * the singular values of [A]. The columns of [U] and [V] are the left and * the right singular vectors of [A], respectively. The matrices [U] and [V] * are computed and stored in the arrays U and V, respectively. The diagonal * of [SIGMA] is computed and stored in the array SVA. * * Arguments * ========= * * JOBA (input) CHARACTER*1 * Specifies the level of accuracy: * = 'C': This option works well (high relative accuracy) if A = B * D, * with well-conditioned B and arbitrary diagonal matrix D. * The accuracy cannot be spoiled by COLUMN scaling. The * accuracy of the computed output depends on the condition of * B, and the procedure aims at the best theoretical accuracy. * The relative error max_{i=1:N}|d sigma_i| / sigma_i is * bounded by f(M,N)*epsilon* cond(B), independent of D. * The input matrix is preprocessed with the QRF with column * pivoting. This initial preprocessing and preconditioning by * a rank revealing QR factorization is common for all values of * JOBA. Additional actions are specified as follows: * = 'E': Computation as with 'C' with an additional estimate of the * condition number of B. It provides a realistic error bound. * = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings * D1, D2, and well-conditioned matrix C, this option gives * higher accuracy than the 'C' option. If the structure of the * input matrix is not known, and relative accuracy is * desirable, then this option is advisable. The input matrix A * is preprocessed with QR factorization with FULL (row and * column) pivoting. * = 'G' Computation as with 'F' with an additional estimate of the * condition number of B, where A=D*B. If A has heavily weighted * rows, then using this condition number gives too pessimistic * error bound. * = 'A': Small singular values are the noise and the matrix is treated * as numerically rank defficient. The error in the computed * singular values is bounded by f(m,n)*epsilon*||A||. * The computed SVD A = U * S * V^t restores A up to * f(m,n)*epsilon*||A||. * This gives the procedure the licence to discard (set to zero) * all singular values below N*epsilon*||A||. * = 'R': Similar as in 'A'. Rank revealing property of the initial * QR factorization is used do reveal (using triangular factor) * a gap sigma_{r+1} < epsilon * sigma_r in which case the * numerical RANK is declared to be r. The SVD is computed with * absolute error bounds, but more accurately than with 'A'. * * JOBU (input) CHARACTER*1 * Specifies whether to compute the columns of U: * = 'U': N columns of U are returned in the array U. * = 'F': full set of M left sing. vectors is returned in the array U. * = 'W': U may be used as workspace of length M*N. See the description * of U. * = 'N': U is not computed. * * JOBV (input) CHARACTER*1 * Specifies whether to compute the matrix V: * = 'V': N columns of V are returned in the array V; Jacobi rotations * are not explicitly accumulated. * = 'J': N columns of V are returned in the array V, but they are * computed as the product of Jacobi rotations. This option is * allowed only if JOBU .NE. 'N', i.e. in computing the full SVD. * = 'W': V may be used as workspace of length N*N. See the description * of V. * = 'N': V is not computed. * * JOBR (input) CHARACTER*1 * Specifies the RANGE for the singular values. Issues the licence to * set to zero small positive singular values if they are outside * specified range. If A .NE. 0 is scaled so that the largest singular * value of c*A is around DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues * the licence to kill columns of A whose norm in c*A is less than * DSQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN, * where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E'). * = 'N': Do not kill small columns of c*A. This option assumes that * BLAS and QR factorizations and triangular solvers are * implemented to work in that range. If the condition of A * is greater than BIG, use DGESVJ. * = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)] * (roughly, as described above). This option is recommended. * ~~~~~~~~~~~~~~~~~~~~~~~~~~~ * For computing the singular values in the FULL range [SFMIN,BIG] * use DGESVJ. * * JOBT (input) CHARACTER*1 * If the matrix is square then the procedure may determine to use * transposed A if A^t seems to be better with respect to convergence. * If the matrix is not square, JOBT is ignored. This is subject to * changes in the future. * The decision is based on two values of entropy over the adjoint * orbit of A^t * A. See the descriptions of WORK(6) and WORK(7). * = 'T': transpose if entropy test indicates possibly faster * convergence of Jacobi process if A^t is taken as input. If A is * replaced with A^t, then the row pivoting is included automatically. * = 'N': do not speculate. * This option can be used to compute only the singular values, or the * full SVD (U, SIGMA and V). For only one set of singular vectors * (U or V), the caller should provide both U and V, as one of the * matrices is used as workspace if the matrix A is transposed. * The implementer can easily remove this constraint and make the * code more complicated. See the descriptions of U and V. * * JOBP (input) CHARACTER*1 * Issues the licence to introduce structured perturbations to drown * denormalized numbers. This licence should be active if the * denormals are poorly implemented, causing slow computation, * especially in cases of fast convergence (!). For details see [1,2]. * For the sake of simplicity, this perturbations are included only * when the full SVD or only the singular values are requested. The * implementer/user can easily add the perturbation for the cases of * computing one set of singular vectors. * = 'P': introduce perturbation * = 'N': do not perturb * * M (input) INTEGER * The number of rows of the input matrix A. M >= 0. * * N (input) INTEGER * The number of columns of the input matrix A. M >= N >= 0. * * A (input/workspace) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the M-by-N matrix A. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * SVA (workspace/output) DOUBLE PRECISION array, dimension (N) * On exit, * - For WORK(1)/WORK(2) = ONE: The singular values of A. During the * computation SVA contains Euclidean column norms of the * iterated matrices in the array A. * - For WORK(1) .NE. WORK(2): The singular values of A are * (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if * sigma_max(A) overflows or if small singular values have been * saved from underflow by scaling the input matrix A. * - If JOBR='R' then some of the singular values may be returned * as exact zeros obtained by "set to zero" because they are * below the numerical rank threshold or are denormalized numbers. * * U (workspace/output) DOUBLE PRECISION array, dimension ( LDU, N ) * If JOBU = 'U', then U contains on exit the M-by-N matrix of * the left singular vectors. * If JOBU = 'F', then U contains on exit the M-by-M matrix of * the left singular vectors, including an ONB * of the orthogonal complement of the Range(A). * If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N), * then U is used as workspace if the procedure * replaces A with A^t. In that case, [V] is computed * in U as left singular vectors of A^t and then * copied back to the V array. This 'W' option is just * a reminder to the caller that in this case U is * reserved as workspace of length N*N. * If JOBU = 'N' U is not referenced. * * LDU (input) INTEGER * The leading dimension of the array U, LDU >= 1. * IF JOBU = 'U' or 'F' or 'W', then LDU >= M. * * V (workspace/output) DOUBLE PRECISION array, dimension ( LDV, N ) * If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of * the right singular vectors; * If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N), * then V is used as workspace if the pprocedure * replaces A with A^t. In that case, [U] is computed * in V as right singular vectors of A^t and then * copied back to the U array. This 'W' option is just * a reminder to the caller that in this case V is * reserved as workspace of length N*N. * If JOBV = 'N' V is not referenced. * * LDV (input) INTEGER * The leading dimension of the array V, LDV >= 1. * If JOBV = 'V' or 'J' or 'W', then LDV >= N. * * WORK (workspace/output) DOUBLE PRECISION array, dimension at least LWORK. * On exit, if N.GT.0 .AND. M.GT.0 (else not referenced), * WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such * that SCALE*SVA(1:N) are the computed singular values * of A. (See the description of SVA().) * WORK(2) = See the description of WORK(1). * WORK(3) = SCONDA is an estimate for the condition number of * column equilibrated A. (If JOBA .EQ. 'E' or 'G') * SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1). * It is computed using DPOCON. It holds * N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA * where R is the triangular factor from the QRF of A. * However, if R is truncated and the numerical rank is * determined to be strictly smaller than N, SCONDA is * returned as -1, thus indicating that the smallest * singular values might be lost. * * If full SVD is needed, the following two condition numbers are * useful for the analysis of the algorithm. They are provied for * a developer/implementer who is familiar with the details of * the method. * * WORK(4) = an estimate of the scaled condition number of the * triangular factor in the first QR factorization. * WORK(5) = an estimate of the scaled condition number of the * triangular factor in the second QR factorization. * The following two parameters are computed if JOBT .EQ. 'T'. * They are provided for a developer/implementer who is familiar * with the details of the method. * * WORK(6) = the entropy of A^t*A :: this is the Shannon entropy * of diag(A^t*A) / Trace(A^t*A) taken as point in the * probability simplex. * WORK(7) = the entropy of A*A^t. * * LWORK (input) INTEGER * Length of WORK to confirm proper allocation of work space. * LWORK depends on the job: * * If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and * -> .. no scaled condition estimate required (JOBE.EQ.'N'): * LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement. * ->> For optimal performance (blocked code) the optimal value * is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal * block size for DGEQP3 and DGEQRF. * In general, optimal LWORK is computed as * LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7). * -> .. an estimate of the scaled condition number of A is * required (JOBA='E', 'G'). In this case, LWORK is the maximum * of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7). * ->> For optimal performance (blocked code) the optimal value * is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7). * In general, the optimal length LWORK is computed as * LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), * N+N*N+LWORK(DPOCON),7). * * If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'), * -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7). * -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7), * where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ, * DORMLQ. In general, the optimal length LWORK is computed as * LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON), * N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)). * * If SIGMA and the left singular vectors are needed * -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7). * -> For optimal performance: * if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7), * if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7), * where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR. * In general, the optimal length LWORK is computed as * LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON), * 2*N+LWORK(DGEQRF), N+LWORK(DORMQR)). * Here LWORK(DORMQR) equals N*NB (for JOBU.EQ.'U') or * M*NB (for JOBU.EQ.'F'). * * If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and * -> if JOBV.EQ.'V' * the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N). * -> if JOBV.EQ.'J' the minimal requirement is * LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6). * -> For optimal performance, LWORK should be additionally * larger than N+M*NB, where NB is the optimal block size * for DORMQR. * * IWORK (workspace/output) INTEGER array, dimension M+3*N. * On exit, * IWORK(1) = the numerical rank determined after the initial * QR factorization with pivoting. See the descriptions * of JOBA and JOBR. * IWORK(2) = the number of the computed nonzero singular values * IWORK(3) = if nonzero, a warning message: * If IWORK(3).EQ.1 then some of the column norms of A * were denormalized floats. The requested high accuracy * is not warranted by the data. * * INFO (output) INTEGER * < 0 : if INFO = -i, then the i-th argument had an illegal value. * = 0 : successfull exit; * > 0 : DGEJSV did not converge in the maximal allowed number * of sweeps. The computed values may be inaccurate. * * Further Details * =============== * * DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses DGEQP3, * DGEQRF, and DGELQF as preprocessors and preconditioners. Optionally, an * additional row pivoting can be used as a preprocessor, which in some * cases results in much higher accuracy. An example is matrix A with the * structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned * diagonal matrices and C is well-conditioned matrix. In that case, complete * pivoting in the first QR factorizations provides accuracy dependent on the * condition number of C, and independent of D1, D2. Such higher accuracy is * not completely understood theoretically, but it works well in practice. * Further, if A can be written as A = B*D, with well-conditioned B and some * diagonal D, then the high accuracy is guaranteed, both theoretically and * in software, independent of D. For more details see [1], [2]. * The computational range for the singular values can be the full range * ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS * & LAPACK routines called by DGEJSV are implemented to work in that range. * If that is not the case, then the restriction for safe computation with * the singular values in the range of normalized IEEE numbers is that the * spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not * overflow. This code (DGEJSV) is best used in this restricted range, * meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are * returned as zeros. See JOBR for details on this. * Further, this implementation is somewhat slower than the one described * in [1,2] due to replacement of some non-LAPACK components, and because * the choice of some tuning parameters in the iterative part (DGESVJ) is * left to the implementer on a particular machine. * The rank revealing QR factorization (in this code: DGEQP3) should be * implemented as in [3]. We have a new version of DGEQP3 under development * that is more robust than the current one in LAPACK, with a cleaner cut in * rank defficient cases. It will be available in the SIGMA library [4]. * If M is much larger than N, it is obvious that the inital QRF with * column pivoting can be preprocessed by the QRF without pivoting. That * well known trick is not used in DGEJSV because in some cases heavy row * weighting can be treated with complete pivoting. The overhead in cases * M much larger than N is then only due to pivoting, but the benefits in * terms of accuracy have prevailed. The implementer/user can incorporate * this extra QRF step easily. The implementer can also improve data movement * (matrix transpose, matrix copy, matrix transposed copy) - this * implementation of DGEJSV uses only the simplest, naive data movement. * * Contributors * * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany) * * References * * [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I. * SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342. * LAPACK Working note 169. * [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II. * SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362. * LAPACK Working note 170. * [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR * factorization software - a case study. * ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28. * LAPACK Working note 176. * [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, * QSVD, (H,K)-SVD computations. * Department of Mathematics, University of Zagreb, 2008. * * Bugs, examples and comments * * Please report all bugs and send interesting examples and/or comments to * drmac@math.hr. Thank you. * * =========================================================================== * * .. Local Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) * .. * .. Local Scalars .. DOUBLE PRECISION AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK, $ CONDR1, CONDR2, ENTRA, ENTRAT, EPSLN, MAXPRJ, SCALEM, $ SCONDA, SFMIN, SMALL, TEMP1, USCAL1, USCAL2, XSC INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LSVEC, $ L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN, $ NOSCAL, ROWPIV, RSVEC, TRANSP * .. * .. Intrinsic Functions .. INTRINSIC DABS, DLOG, DMAX1, DMIN1, DBLE, $ MAX0, MIN0, IDNINT, DSIGN, DSQRT * .. * .. External Functions .. DOUBLE PRECISION DLAMCH, DNRM2 INTEGER IDAMAX LOGICAL LSAME EXTERNAL IDAMAX, LSAME, DLAMCH, DNRM2 * .. * .. External Subroutines .. EXTERNAL DCOPY, DGELQF, DGEQP3, DGEQRF, DLACPY, DLASCL, $ DLASET, DLASSQ, DLASWP, DORGQR, DORMLQ, $ DORMQR, DPOCON, DSCAL, DSWAP, DTRSM, XERBLA * EXTERNAL DGESVJ * .. * * Test the input arguments * LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' ) JRACC = LSAME( JOBV, 'J' ) RSVEC = LSAME( JOBV, 'V' ) .OR. JRACC ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' ) L2RANK = LSAME( JOBA, 'R' ) L2ABER = LSAME( JOBA, 'A' ) ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' ) L2TRAN = LSAME( JOBT, 'T' ) L2KILL = LSAME( JOBR, 'R' ) DEFR = LSAME( JOBR, 'N' ) L2PERT = LSAME( JOBP, 'P' ) * IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR. $ ERREST .OR. LSAME( JOBA, 'C' ) )) THEN INFO = - 1 ELSE IF ( .NOT.( LSVEC .OR. LSAME( JOBU, 'N' ) .OR. $ LSAME( JOBU, 'W' )) ) THEN INFO = - 2 ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR. $ LSAME( JOBV, 'W' )) .OR. ( JRACC .AND. (.NOT.LSVEC) ) ) THEN INFO = - 3 ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) ) THEN INFO = - 4 ELSE IF ( .NOT. ( L2TRAN .OR. LSAME( JOBT, 'N' ) ) ) THEN INFO = - 5 ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN INFO = - 6 ELSE IF ( M .LT. 0 ) THEN INFO = - 7 ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN INFO = - 8 ELSE IF ( LDA .LT. M ) THEN INFO = - 10 ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN INFO = - 13 ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN INFO = - 14 ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND. & (LWORK .LT. MAX0(7,4*N+1,2*M+N))) .OR. & (.NOT.(LSVEC .OR. RSVEC) .AND. ERREST .AND. & (LWORK .LT. MAX0(7,4*N+N*N,2*M+N))) .OR. & (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX0(7,2*M+N,4*N+1))) & .OR. & (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX0(7,2*M+N,4*N+1))) & .OR. & (LSVEC .AND. RSVEC .AND. (.NOT.JRACC) .AND. & (LWORK.LT.MAX0(2*M+N,6*N+2*N*N))) & .OR. (LSVEC .AND. RSVEC .AND. JRACC .AND. & LWORK.LT.MAX0(2*M+N,4*N+N*N,2*N+N*N+6))) & THEN INFO = - 17 ELSE * #:) INFO = 0 END IF * IF ( INFO .NE. 0 ) THEN * #:( CALL XERBLA( 'DGEJSV', - INFO ) RETURN END IF * * Quick return for void matrix (Y3K safe) * #:) IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) RETURN * * Determine whether the matrix U should be M x N or M x M * IF ( LSVEC ) THEN N1 = N IF ( LSAME( JOBU, 'F' ) ) N1 = M END IF * * Set numerical parameters * *! NOTE: Make sure DLAMCH() does not fail on the target architecture. * EPSLN = DLAMCH('Epsilon') SFMIN = DLAMCH('SafeMinimum') SMALL = SFMIN / EPSLN BIG = DLAMCH('O') * BIG = ONE / SFMIN * * Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N * *(!) If necessary, scale SVA() to protect the largest norm from * overflow. It is possible that this scaling pushes the smallest * column norm left from the underflow threshold (extreme case). * SCALEM = ONE / DSQRT(DBLE(M)*DBLE(N)) NOSCAL = .TRUE. GOSCAL = .TRUE. DO 1874 p = 1, N AAPP = ZERO AAQQ = ONE CALL DLASSQ( M, A(1,p), 1, AAPP, AAQQ ) IF ( AAPP .GT. BIG ) THEN INFO = - 9 CALL XERBLA( 'DGEJSV', -INFO ) RETURN END IF AAQQ = DSQRT(AAQQ) IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL ) THEN SVA(p) = AAPP * AAQQ ELSE NOSCAL = .FALSE. SVA(p) = AAPP * ( AAQQ * SCALEM ) IF ( GOSCAL ) THEN GOSCAL = .FALSE. CALL DSCAL( p-1, SCALEM, SVA, 1 ) END IF END IF 1874 CONTINUE * IF ( NOSCAL ) SCALEM = ONE * AAPP = ZERO AAQQ = BIG DO 4781 p = 1, N AAPP = DMAX1( AAPP, SVA(p) ) IF ( SVA(p) .NE. ZERO ) AAQQ = DMIN1( AAQQ, SVA(p) ) 4781 CONTINUE * * Quick return for zero M x N matrix * #:) IF ( AAPP .EQ. ZERO ) THEN IF ( LSVEC ) CALL DLASET( 'G', M, N1, ZERO, ONE, U, LDU ) IF ( RSVEC ) CALL DLASET( 'G', N, N, ZERO, ONE, V, LDV ) WORK(1) = ONE WORK(2) = ONE IF ( ERREST ) WORK(3) = ONE IF ( LSVEC .AND. RSVEC ) THEN WORK(4) = ONE WORK(5) = ONE END IF IF ( L2TRAN ) THEN WORK(6) = ZERO WORK(7) = ZERO END IF IWORK(1) = 0 IWORK(2) = 0 IWORK(3) = 0 RETURN END IF * * Issue warning if denormalized column norms detected. Override the * high relative accuracy request. Issue licence to kill columns * (set them to zero) whose norm is less than sigma_max / BIG (roughly). * #:( WARNING = 0 IF ( AAQQ .LE. SFMIN ) THEN L2RANK = .TRUE. L2KILL = .TRUE. WARNING = 1 END IF * * Quick return for one-column matrix * #:) IF ( N .EQ. 1 ) THEN * IF ( LSVEC ) THEN CALL DLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR ) CALL DLACPY( 'A', M, 1, A, LDA, U, LDU ) * computing all M left singular vectors of the M x 1 matrix IF ( N1 .NE. N ) THEN CALL DGEQRF( M, N, U,LDU, WORK, WORK(N+1),LWORK-N,IERR ) CALL DORGQR( M,N1,1, U,LDU,WORK,WORK(N+1),LWORK-N,IERR ) CALL DCOPY( M, A(1,1), 1, U(1,1), 1 ) END IF END IF IF ( RSVEC ) THEN V(1,1) = ONE END IF IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN SVA(1) = SVA(1) / SCALEM SCALEM = ONE END IF WORK(1) = ONE / SCALEM WORK(2) = ONE IF ( SVA(1) .NE. ZERO ) THEN IWORK(1) = 1 IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN IWORK(2) = 1 ELSE IWORK(2) = 0 END IF ELSE IWORK(1) = 0 IWORK(2) = 0 END IF IF ( ERREST ) WORK(3) = ONE IF ( LSVEC .AND. RSVEC ) THEN WORK(4) = ONE WORK(5) = ONE END IF IF ( L2TRAN ) THEN WORK(6) = ZERO WORK(7) = ZERO END IF RETURN * END IF * TRANSP = .FALSE. L2TRAN = L2TRAN .AND. ( M .EQ. N ) * AATMAX = -ONE AATMIN = BIG IF ( ROWPIV .OR. L2TRAN ) THEN * * Compute the row norms, needed to determine row pivoting sequence * (in the case of heavily row weighted A, row pivoting is strongly * advised) and to collect information needed to compare the * structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.). * IF ( L2TRAN ) THEN DO 1950 p = 1, M XSC = ZERO TEMP1 = ONE CALL DLASSQ( N, A(p,1), LDA, XSC, TEMP1 ) * DLASSQ gets both the ell_2 and the ell_infinity norm * in one pass through the vector WORK(M+N+p) = XSC * SCALEM WORK(N+p) = XSC * (SCALEM*DSQRT(TEMP1)) AATMAX = DMAX1( AATMAX, WORK(N+p) ) IF (WORK(N+p) .NE. ZERO) AATMIN = DMIN1(AATMIN,WORK(N+p)) 1950 CONTINUE ELSE DO 1904 p = 1, M WORK(M+N+p) = SCALEM*DABS( A(p,IDAMAX(N,A(p,1),LDA)) ) AATMAX = DMAX1( AATMAX, WORK(M+N+p) ) AATMIN = DMIN1( AATMIN, WORK(M+N+p) ) 1904 CONTINUE END IF * END IF * * For square matrix A try to determine whether A^t would be better * input for the preconditioned Jacobi SVD, with faster convergence. * The decision is based on an O(N) function of the vector of column * and row norms of A, based on the Shannon entropy. This should give * the right choice in most cases when the difference actually matters. * It may fail and pick the slower converging side. * ENTRA = ZERO ENTRAT = ZERO IF ( L2TRAN ) THEN * XSC = ZERO TEMP1 = ONE CALL DLASSQ( N, SVA, 1, XSC, TEMP1 ) TEMP1 = ONE / TEMP1 * ENTRA = ZERO DO 1113 p = 1, N BIG1 = ( ( SVA(p) / XSC )**2 ) * TEMP1 IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * DLOG(BIG1) 1113 CONTINUE ENTRA = - ENTRA / DLOG(DBLE(N)) * * Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex. * It is derived from the diagonal of A^t * A. Do the same with the * diagonal of A * A^t, compute the entropy of the corresponding * probability distribution. Note that A * A^t and A^t * A have the * same trace. * ENTRAT = ZERO DO 1114 p = N+1, N+M BIG1 = ( ( WORK(p) / XSC )**2 ) * TEMP1 IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * DLOG(BIG1) 1114 CONTINUE ENTRAT = - ENTRAT / DLOG(DBLE(M)) * * Analyze the entropies and decide A or A^t. Smaller entropy * usually means better input for the algorithm. * TRANSP = ( ENTRAT .LT. ENTRA ) * * If A^t is better than A, transpose A. * IF ( TRANSP ) THEN * In an optimal implementation, this trivial transpose * should be replaced with faster transpose. DO 1115 p = 1, N - 1 DO 1116 q = p + 1, N TEMP1 = A(q,p) A(q,p) = A(p,q) A(p,q) = TEMP1 1116 CONTINUE 1115 CONTINUE DO 1117 p = 1, N WORK(M+N+p) = SVA(p) SVA(p) = WORK(N+p) 1117 CONTINUE TEMP1 = AAPP AAPP = AATMAX AATMAX = TEMP1 TEMP1 = AAQQ AAQQ = AATMIN AATMIN = TEMP1 KILL = LSVEC LSVEC = RSVEC RSVEC = KILL IF ( LSVEC ) N1 = N * ROWPIV = .TRUE. END IF * END IF * END IF L2TRAN * * Scale the matrix so that its maximal singular value remains less * than DSQRT(BIG) -- the matrix is scaled so that its maximal column * has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep * DSQRT(BIG) instead of BIG is the fact that DGEJSV uses LAPACK and * BLAS routines that, in some implementations, are not capable of * working in the full interval [SFMIN,BIG] and that they may provoke * overflows in the intermediate results. If the singular values spread * from SFMIN to BIG, then DGESVJ will compute them. So, in that case, * one should use DGESVJ instead of DGEJSV. * BIG1 = DSQRT( BIG ) TEMP1 = DSQRT( BIG / DBLE(N) ) * CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR ) IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN AAQQ = ( AAQQ / AAPP ) * TEMP1 ELSE AAQQ = ( AAQQ * TEMP1 ) / AAPP END IF TEMP1 = TEMP1 * SCALEM CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR ) * * To undo scaling at the end of this procedure, multiply the * computed singular values with USCAL2 / USCAL1. * USCAL1 = TEMP1 USCAL2 = AAPP * IF ( L2KILL ) THEN * L2KILL enforces computation of nonzero singular values in * the restricted range of condition number of the initial A, * sigma_max(A) / sigma_min(A) approx. DSQRT(BIG)/DSQRT(SFMIN). XSC = DSQRT( SFMIN ) ELSE XSC = SMALL * * Now, if the condition number of A is too big, * sigma_max(A) / sigma_min(A) .GT. DSQRT(BIG/N) * EPSLN / SFMIN, * as a precaution measure, the full SVD is computed using DGESVJ * with accumulated Jacobi rotations. This provides numerically * more robust computation, at the cost of slightly increased run * time. Depending on the concrete implementation of BLAS and LAPACK * (i.e. how they behave in presence of extreme ill-conditioning) the * implementor may decide to remove this switch. IF ( ( AAQQ.LT.DSQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN JRACC = .TRUE. END IF * END IF IF ( AAQQ .LT. XSC ) THEN DO 700 p = 1, N IF ( SVA(p) .LT. XSC ) THEN CALL DLASET( 'A', M, 1, ZERO, ZERO, A(1,p), LDA ) SVA(p) = ZERO END IF 700 CONTINUE END IF * * Preconditioning using QR factorization with pivoting * IF ( ROWPIV ) THEN * Optional row permutation (Bjoerck row pivoting): * A result by Cox and Higham shows that the Bjoerck's * row pivoting combined with standard column pivoting * has similar effect as Powell-Reid complete pivoting. * The ell-infinity norms of A are made nonincreasing. DO 1952 p = 1, M - 1 q = IDAMAX( M-p+1, WORK(M+N+p), 1 ) + p - 1 IWORK(2*N+p) = q IF ( p .NE. q ) THEN TEMP1 = WORK(M+N+p) WORK(M+N+p) = WORK(M+N+q) WORK(M+N+q) = TEMP1 END IF 1952 CONTINUE CALL DLASWP( N, A, LDA, 1, M-1, IWORK(2*N+1), 1 ) END IF * * End of the preparation phase (scaling, optional sorting and * transposing, optional flushing of small columns). * * Preconditioning * * If the full SVD is needed, the right singular vectors are computed * from a matrix equation, and for that we need theoretical analysis * of the Businger-Golub pivoting. So we use DGEQP3 as the first RR QRF. * In all other cases the first RR QRF can be chosen by other criteria * (eg speed by replacing global with restricted window pivoting, such * as in SGEQPX from TOMS # 782). Good results will be obtained using * SGEQPX with properly (!) chosen numerical parameters. * Any improvement of DGEQP3 improves overal performance of DGEJSV. * * A * P1 = Q1 * [ R1^t 0]^t: DO 1963 p = 1, N * .. all columns are free columns IWORK(p) = 0 1963 CONTINUE CALL DGEQP3( M,N,A,LDA, IWORK,WORK, WORK(N+1),LWORK-N, IERR ) * * The upper triangular matrix R1 from the first QRF is inspected for * rank deficiency and possibilities for deflation, or possible * ill-conditioning. Depending on the user specified flag L2RANK, * the procedure explores possibilities to reduce the numerical * rank by inspecting the computed upper triangular factor. If * L2RANK or L2ABER are up, then DGEJSV will compute the SVD of * A + dA, where ||dA|| <= f(M,N)*EPSLN. * NR = 1 IF ( L2ABER ) THEN * Standard absolute error bound suffices. All sigma_i with * sigma_i < N*EPSLN*||A|| are flushed to zero. This is an * agressive enforcement of lower numerical rank by introducing a * backward error of the order of N*EPSLN*||A||. TEMP1 = DSQRT(DBLE(N))*EPSLN DO 3001 p = 2, N IF ( DABS(A(p,p)) .GE. (TEMP1*DABS(A(1,1))) ) THEN NR = NR + 1 ELSE GO TO 3002 END IF 3001 CONTINUE 3002 CONTINUE ELSE IF ( L2RANK ) THEN * .. similarly as above, only slightly more gentle (less agressive). * Sudden drop on the diagonal of R1 is used as the criterion for * close-to-rank-defficient. TEMP1 = DSQRT(SFMIN) DO 3401 p = 2, N IF ( ( DABS(A(p,p)) .LT. (EPSLN*DABS(A(p-1,p-1))) ) .OR. $ ( DABS(A(p,p)) .LT. SMALL ) .OR. $ ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402 NR = NR + 1 3401 CONTINUE 3402 CONTINUE * ELSE * The goal is high relative accuracy. However, if the matrix * has high scaled condition number the relative accuracy is in * general not feasible. Later on, a condition number estimator * will be deployed to estimate the scaled condition number. * Here we just remove the underflowed part of the triangular * factor. This prevents the situation in which the code is * working hard to get the accuracy not warranted by the data. TEMP1 = DSQRT(SFMIN) DO 3301 p = 2, N IF ( ( DABS(A(p,p)) .LT. SMALL ) .OR. $ ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302 NR = NR + 1 3301 CONTINUE 3302 CONTINUE * END IF * ALMORT = .FALSE. IF ( NR .EQ. N ) THEN MAXPRJ = ONE DO 3051 p = 2, N TEMP1 = DABS(A(p,p)) / SVA(IWORK(p)) MAXPRJ = DMIN1( MAXPRJ, TEMP1 ) 3051 CONTINUE IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE. END IF * * SCONDA = - ONE CONDR1 = - ONE CONDR2 = - ONE * IF ( ERREST ) THEN IF ( N .EQ. NR ) THEN IF ( RSVEC ) THEN * .. V is available as workspace CALL DLACPY( 'U', N, N, A, LDA, V, LDV ) DO 3053 p = 1, N TEMP1 = SVA(IWORK(p)) CALL DSCAL( p, ONE/TEMP1, V(1,p), 1 ) 3053 CONTINUE CALL DPOCON( 'U', N, V, LDV, ONE, TEMP1, $ WORK(N+1), IWORK(2*N+M+1), IERR ) ELSE IF ( LSVEC ) THEN * .. U is available as workspace CALL DLACPY( 'U', N, N, A, LDA, U, LDU ) DO 3054 p = 1, N TEMP1 = SVA(IWORK(p)) CALL DSCAL( p, ONE/TEMP1, U(1,p), 1 ) 3054 CONTINUE CALL DPOCON( 'U', N, U, LDU, ONE, TEMP1, $ WORK(N+1), IWORK(2*N+M+1), IERR ) ELSE CALL DLACPY( 'U', N, N, A, LDA, WORK(N+1), N ) DO 3052 p = 1, N TEMP1 = SVA(IWORK(p)) CALL DSCAL( p, ONE/TEMP1, WORK(N+(p-1)*N+1), 1 ) 3052 CONTINUE * .. the columns of R are scaled to have unit Euclidean lengths. CALL DPOCON( 'U', N, WORK(N+1), N, ONE, TEMP1, $ WORK(N+N*N+1), IWORK(2*N+M+1), IERR ) END IF SCONDA = ONE / DSQRT(TEMP1) * SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1). * N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA ELSE SCONDA = - ONE END IF END IF * L2PERT = L2PERT .AND. ( DABS( A(1,1)/A(NR,NR) ) .GT. DSQRT(BIG1) ) * If there is no violent scaling, artificial perturbation is not needed. * * Phase 3: * IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN * * Singular Values only * * .. transpose A(1:NR,1:N) DO 1946 p = 1, MIN0( N-1, NR ) CALL DCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 ) 1946 CONTINUE * * The following two DO-loops introduce small relative perturbation * into the strict upper triangle of the lower triangular matrix. * Small entries below the main diagonal are also changed. * This modification is useful if the computing environment does not * provide/allow FLUSH TO ZERO underflow, for it prevents many * annoying denormalized numbers in case of strongly scaled matrices. * The perturbation is structured so that it does not introduce any * new perturbation of the singular values, and it does not destroy * the job done by the preconditioner. * The licence for this perturbation is in the variable L2PERT, which * should be .FALSE. if FLUSH TO ZERO underflow is active. * IF ( .NOT. ALMORT ) THEN * IF ( L2PERT ) THEN * XSC = DSQRT(SMALL) XSC = EPSLN / DBLE(N) DO 4947 q = 1, NR TEMP1 = XSC*DABS(A(q,q)) DO 4949 p = 1, N IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) ) $ .OR. ( p .LT. q ) ) $ A(p,q) = DSIGN( TEMP1, A(p,q) ) 4949 CONTINUE 4947 CONTINUE ELSE CALL DLASET( 'U', NR-1,NR-1, ZERO,ZERO, A(1,2),LDA ) END IF * * .. second preconditioning using the QR factorization * CALL DGEQRF( N,NR, A,LDA, WORK, WORK(N+1),LWORK-N, IERR ) * * .. and transpose upper to lower triangular DO 1948 p = 1, NR - 1 CALL DCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 ) 1948 CONTINUE * END IF * * Row-cyclic Jacobi SVD algorithm with column pivoting * * .. again some perturbation (a "background noise") is added * to drown denormals IF ( L2PERT ) THEN * XSC = DSQRT(SMALL) XSC = EPSLN / DBLE(N) DO 1947 q = 1, NR TEMP1 = XSC*DABS(A(q,q)) DO 1949 p = 1, NR IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) ) $ .OR. ( p .LT. q ) ) $ A(p,q) = DSIGN( TEMP1, A(p,q) ) 1949 CONTINUE 1947 CONTINUE ELSE CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, A(1,2), LDA ) END IF * * .. and one-sided Jacobi rotations are started on a lower * triangular matrix (plus perturbation which is ignored in * the part which destroys triangular form (confusing?!)) * CALL DGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA, $ N, V, LDV, WORK, LWORK, INFO ) * SCALEM = WORK(1) NUMRANK = IDNINT(WORK(2)) * * ELSE IF ( RSVEC .AND. ( .NOT. LSVEC ) ) THEN * * -> Singular Values and Right Singular Vectors <- * IF ( ALMORT ) THEN * * .. in this case NR equals N DO 1998 p = 1, NR CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 ) 1998 CONTINUE CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV ) * CALL DGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA, $ WORK, LWORK, INFO ) SCALEM = WORK(1) NUMRANK = IDNINT(WORK(2)) ELSE * * .. two more QR factorizations ( one QRF is not enough, two require * accumulated product of Jacobi rotations, three are perfect ) * CALL DLASET( 'Lower', NR-1, NR-1, ZERO, ZERO, A(2,1), LDA ) CALL DGELQF( NR, N, A, LDA, WORK, WORK(N+1), LWORK-N, IERR) CALL DLACPY( 'Lower', NR, NR, A, LDA, V, LDV ) CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV ) CALL DGEQRF( NR, NR, V, LDV, WORK(N+1), WORK(2*N+1), $ LWORK-2*N, IERR ) DO 8998 p = 1, NR CALL DCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 ) 8998 CONTINUE CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV ) * CALL DGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U, $ LDU, WORK(N+1), LWORK, INFO ) SCALEM = WORK(N+1) NUMRANK = IDNINT(WORK(N+2)) IF ( NR .LT. N ) THEN CALL DLASET( 'A',N-NR, NR, ZERO,ZERO, V(NR+1,1), LDV ) CALL DLASET( 'A',NR, N-NR, ZERO,ZERO, V(1,NR+1), LDV ) CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE, V(NR+1,NR+1), LDV ) END IF * CALL DORMLQ( 'Left', 'Transpose', N, N, NR, A, LDA, WORK, $ V, LDV, WORK(N+1), LWORK-N, IERR ) * END IF * DO 8991 p = 1, N CALL DCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA ) 8991 CONTINUE CALL DLACPY( 'All', N, N, A, LDA, V, LDV ) * IF ( TRANSP ) THEN CALL DLACPY( 'All', N, N, V, LDV, U, LDU ) END IF * ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN * * .. Singular Values and Left Singular Vectors .. * * .. second preconditioning step to avoid need to accumulate * Jacobi rotations in the Jacobi iterations. DO 1965 p = 1, NR CALL DCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 ) 1965 CONTINUE CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU ) * CALL DGEQRF( N, NR, U, LDU, WORK(N+1), WORK(2*N+1), $ LWORK-2*N, IERR ) * DO 1967 p = 1, NR - 1 CALL DCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 ) 1967 CONTINUE CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU ) * CALL DGESVJ( 'Lower', 'U', 'N', NR,NR, U, LDU, SVA, NR, A, $ LDA, WORK(N+1), LWORK-N, INFO ) SCALEM = WORK(N+1) NUMRANK = IDNINT(WORK(N+2)) * IF ( NR .LT. M ) THEN CALL DLASET( 'A', M-NR, NR,ZERO, ZERO, U(NR+1,1), LDU ) IF ( NR .LT. N1 ) THEN CALL DLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1), LDU ) CALL DLASET( 'A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1), LDU ) END IF END IF * CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U, $ LDU, WORK(N+1), LWORK-N, IERR ) * IF ( ROWPIV ) $ CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 ) * DO 1974 p = 1, N1 XSC = ONE / DNRM2( M, U(1,p), 1 ) CALL DSCAL( M, XSC, U(1,p), 1 ) 1974 CONTINUE * IF ( TRANSP ) THEN CALL DLACPY( 'All', N, N, U, LDU, V, LDV ) END IF * ELSE * * .. Full SVD .. * IF ( .NOT. JRACC ) THEN * IF ( .NOT. ALMORT ) THEN * * Second Preconditioning Step (QRF [with pivoting]) * Note that the composition of TRANSPOSE, QRF and TRANSPOSE is * equivalent to an LQF CALL. Since in many libraries the QRF * seems to be better optimized than the LQF, we do explicit * transpose and use the QRF. This is subject to changes in an * optimized implementation of DGEJSV. * DO 1968 p = 1, NR CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 ) 1968 CONTINUE * * .. the following two loops perturb small entries to avoid * denormals in the second QR factorization, where they are * as good as zeros. This is done to avoid painfully slow * computation with denormals. The relative size of the perturbation * is a parameter that can be changed by the implementer. * This perturbation device will be obsolete on machines with * properly implemented arithmetic. * To switch it off, set L2PERT=.FALSE. To remove it from the * code, remove the action under L2PERT=.TRUE., leave the ELSE part. * The following two loops should be blocked and fused with the * transposed copy above. * IF ( L2PERT ) THEN XSC = DSQRT(SMALL) DO 2969 q = 1, NR TEMP1 = XSC*DABS( V(q,q) ) DO 2968 p = 1, N IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 ) $ .OR. ( p .LT. q ) ) $ V(p,q) = DSIGN( TEMP1, V(p,q) ) IF ( p .LT. q ) V(p,q) = - V(p,q) 2968 CONTINUE 2969 CONTINUE ELSE CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV ) END IF * * Estimate the row scaled condition number of R1 * (If R1 is rectangular, N > NR, then the condition number * of the leading NR x NR submatrix is estimated.) * CALL DLACPY( 'L', NR, NR, V, LDV, WORK(2*N+1), NR ) DO 3950 p = 1, NR TEMP1 = DNRM2(NR-p+1,WORK(2*N+(p-1)*NR+p),1) CALL DSCAL(NR-p+1,ONE/TEMP1,WORK(2*N+(p-1)*NR+p),1) 3950 CONTINUE CALL DPOCON('Lower',NR,WORK(2*N+1),NR,ONE,TEMP1, $ WORK(2*N+NR*NR+1),IWORK(M+2*N+1),IERR) CONDR1 = ONE / DSQRT(TEMP1) * .. here need a second oppinion on the condition number * .. then assume worst case scenario * R1 is OK for inverse <=> CONDR1 .LT. DBLE(N) * more conservative <=> CONDR1 .LT. DSQRT(DBLE(N)) * COND_OK = DSQRT(DBLE(NR)) *[TP] COND_OK is a tuning parameter. IF ( CONDR1 .LT. COND_OK ) THEN * .. the second QRF without pivoting. Note: in an optimized * implementation, this QRF should be implemented as the QRF * of a lower triangular matrix. * R1^t = Q2 * R2 CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1), $ LWORK-2*N, IERR ) * IF ( L2PERT ) THEN XSC = DSQRT(SMALL)/EPSLN DO 3959 p = 2, NR DO 3958 q = 1, p - 1 TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q))) IF ( DABS(V(q,p)) .LE. TEMP1 ) $ V(q,p) = DSIGN( TEMP1, V(q,p) ) 3958 CONTINUE 3959 CONTINUE END IF * IF ( NR .NE. N ) $ CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N ) * .. save ... * * .. this transposed copy should be better than naive DO 1969 p = 1, NR - 1 CALL DCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 ) 1969 CONTINUE * CONDR2 = CONDR1 * ELSE * * .. ill-conditioned case: second QRF with pivoting * Note that windowed pivoting would be equaly good * numerically, and more run-time efficient. So, in * an optimal implementation, the next call to DGEQP3 * should be replaced with eg. CALL SGEQPX (ACM TOMS #782) * with properly (carefully) chosen parameters. * * R1^t * P2 = Q2 * R2 DO 3003 p = 1, NR IWORK(N+p) = 0 3003 CONTINUE CALL DGEQP3( N, NR, V, LDV, IWORK(N+1), WORK(N+1), $ WORK(2*N+1), LWORK-2*N, IERR ) ** CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1), ** $ LWORK-2*N, IERR ) IF ( L2PERT ) THEN XSC = DSQRT(SMALL) DO 3969 p = 2, NR DO 3968 q = 1, p - 1 TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q))) IF ( DABS(V(q,p)) .LE. TEMP1 ) $ V(q,p) = DSIGN( TEMP1, V(q,p) ) 3968 CONTINUE 3969 CONTINUE END IF * CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N ) * IF ( L2PERT ) THEN XSC = DSQRT(SMALL) DO 8970 p = 2, NR DO 8971 q = 1, p - 1 TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q))) V(p,q) = - DSIGN( TEMP1, V(q,p) ) 8971 CONTINUE 8970 CONTINUE ELSE CALL DLASET( 'L',NR-1,NR-1,ZERO,ZERO,V(2,1),LDV ) END IF * Now, compute R2 = L3 * Q3, the LQ factorization. CALL DGELQF( NR, NR, V, LDV, WORK(2*N+N*NR+1), $ WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR ) * .. and estimate the condition number CALL DLACPY( 'L',NR,NR,V,LDV,WORK(2*N+N*NR+NR+1),NR ) DO 4950 p = 1, NR TEMP1 = DNRM2( p, WORK(2*N+N*NR+NR+p), NR ) CALL DSCAL( p, ONE/TEMP1, WORK(2*N+N*NR+NR+p), NR ) 4950 CONTINUE CALL DPOCON( 'L',NR,WORK(2*N+N*NR+NR+1),NR,ONE,TEMP1, $ WORK(2*N+N*NR+NR+NR*NR+1),IWORK(M+2*N+1),IERR ) CONDR2 = ONE / DSQRT(TEMP1) * IF ( CONDR2 .GE. COND_OK ) THEN * .. save the Householder vectors used for Q3 * (this overwrittes the copy of R2, as it will not be * needed in this branch, but it does not overwritte the * Huseholder vectors of Q2.). CALL DLACPY( 'U', NR, NR, V, LDV, WORK(2*N+1), N ) * .. and the rest of the information on Q3 is in * WORK(2*N+N*NR+1:2*N+N*NR+N) END IF * END IF * IF ( L2PERT ) THEN XSC = DSQRT(SMALL) DO 4968 q = 2, NR TEMP1 = XSC * V(q,q) DO 4969 p = 1, q - 1 * V(p,q) = - DSIGN( TEMP1, V(q,p) ) V(p,q) = - DSIGN( TEMP1, V(p,q) ) 4969 CONTINUE 4968 CONTINUE ELSE CALL DLASET( 'U', NR-1,NR-1, ZERO,ZERO, V(1,2), LDV ) END IF * * Second preconditioning finished; continue with Jacobi SVD * The input matrix is lower trinagular. * * Recover the right singular vectors as solution of a well * conditioned triangular matrix equation. * IF ( CONDR1 .LT. COND_OK ) THEN * CALL DGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U, $ LDU,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,INFO ) SCALEM = WORK(2*N+N*NR+NR+1) NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2)) DO 3970 p = 1, NR CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 ) CALL DSCAL( NR, SVA(p), V(1,p), 1 ) 3970 CONTINUE * .. pick the right matrix equation and solve it * IF ( NR .EQ. N ) THEN * :)) .. best case, R1 is inverted. The solution of this matrix * equation is Q2*V2 = the product of the Jacobi rotations * used in DGESVJ, premultiplied with the orthogonal matrix * from the second QR factorization. CALL DTRSM( 'L','U','N','N', NR,NR,ONE, A,LDA, V,LDV ) ELSE * .. R1 is well conditioned, but non-square. Transpose(R2) * is inverted to get the product of the Jacobi rotations * used in DGESVJ. The Q-factor from the second QR * factorization is then built in explicitly. CALL DTRSM('L','U','T','N',NR,NR,ONE,WORK(2*N+1), $ N,V,LDV) IF ( NR .LT. N ) THEN CALL DLASET('A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV) CALL DLASET('A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV) CALL DLASET('A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV) END IF CALL DORMQR('L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1), $ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR) END IF * ELSE IF ( CONDR2 .LT. COND_OK ) THEN * * :) .. the input matrix A is very likely a relative of * the Kahan matrix :) * The matrix R2 is inverted. The solution of the matrix equation * is Q3^T*V3 = the product of the Jacobi rotations (appplied to * the lower triangular L3 from the LQ factorization of * R2=L3*Q3), pre-multiplied with the transposed Q3. CALL DGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U, $ LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO ) SCALEM = WORK(2*N+N*NR+NR+1) NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2)) DO 3870 p = 1, NR CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 ) CALL DSCAL( NR, SVA(p), U(1,p), 1 ) 3870 CONTINUE CALL DTRSM('L','U','N','N',NR,NR,ONE,WORK(2*N+1),N,U,LDU) * .. apply the permutation from the second QR factorization DO 873 q = 1, NR DO 872 p = 1, NR WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q) 872 CONTINUE DO 874 p = 1, NR U(p,q) = WORK(2*N+N*NR+NR+p) 874 CONTINUE 873 CONTINUE IF ( NR .LT. N ) THEN CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV ) CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV ) CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV ) END IF CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1), $ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR ) ELSE * Last line of defense. * #:( This is a rather pathological case: no scaled condition * improvement after two pivoted QR factorizations. Other * possibility is that the rank revealing QR factorization * or the condition estimator has failed, or the COND_OK * is set very close to ONE (which is unnecessary). Normally, * this branch should never be executed, but in rare cases of * failure of the RRQR or condition estimator, the last line of * defense ensures that DGEJSV completes the task. * Compute the full SVD of L3 using DGESVJ with explicit * accumulation of Jacobi rotations. CALL DGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U, $ LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO ) SCALEM = WORK(2*N+N*NR+NR+1) NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2)) IF ( NR .LT. N ) THEN CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV ) CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV ) CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV ) END IF CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1), $ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR ) * CALL DORMLQ( 'L', 'T', NR, NR, NR, WORK(2*N+1), N, $ WORK(2*N+N*NR+1), U, LDU, WORK(2*N+N*NR+NR+1), $ LWORK-2*N-N*NR-NR, IERR ) DO 773 q = 1, NR DO 772 p = 1, NR WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q) 772 CONTINUE DO 774 p = 1, NR U(p,q) = WORK(2*N+N*NR+NR+p) 774 CONTINUE 773 CONTINUE * END IF * * Permute the rows of V using the (column) permutation from the * first QRF. Also, scale the columns to make them unit in * Euclidean norm. This applies to all cases. * TEMP1 = DSQRT(DBLE(N)) * EPSLN DO 1972 q = 1, N DO 972 p = 1, N WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q) 972 CONTINUE DO 973 p = 1, N V(p,q) = WORK(2*N+N*NR+NR+p) 973 CONTINUE XSC = ONE / DNRM2( N, V(1,q), 1 ) IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) ) $ CALL DSCAL( N, XSC, V(1,q), 1 ) 1972 CONTINUE * At this moment, V contains the right singular vectors of A. * Next, assemble the left singular vector matrix U (M x N). IF ( NR .LT. M ) THEN CALL DLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU ) IF ( NR .LT. N1 ) THEN CALL DLASET('A',NR,N1-NR,ZERO,ZERO,U(1,NR+1),LDU) CALL DLASET('A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1),LDU) END IF END IF * * The Q matrix from the first QRF is built into the left singular * matrix U. This applies to all cases. * CALL DORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, WORK, U, $ LDU, WORK(N+1), LWORK-N, IERR ) * The columns of U are normalized. The cost is O(M*N) flops. TEMP1 = DSQRT(DBLE(M)) * EPSLN DO 1973 p = 1, NR XSC = ONE / DNRM2( M, U(1,p), 1 ) IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) ) $ CALL DSCAL( M, XSC, U(1,p), 1 ) 1973 CONTINUE * * If the initial QRF is computed with row pivoting, the left * singular vectors must be adjusted. * IF ( ROWPIV ) $ CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 ) * ELSE * * .. the initial matrix A has almost orthogonal columns and * the second QRF is not needed * CALL DLACPY( 'Upper', N, N, A, LDA, WORK(N+1), N ) IF ( L2PERT ) THEN XSC = DSQRT(SMALL) DO 5970 p = 2, N TEMP1 = XSC * WORK( N + (p-1)*N + p ) DO 5971 q = 1, p - 1 WORK(N+(q-1)*N+p)=-DSIGN(TEMP1,WORK(N+(p-1)*N+q)) 5971 CONTINUE 5970 CONTINUE ELSE CALL DLASET( 'Lower',N-1,N-1,ZERO,ZERO,WORK(N+2),N ) END IF * CALL DGESVJ( 'Upper', 'U', 'N', N, N, WORK(N+1), N, SVA, $ N, U, LDU, WORK(N+N*N+1), LWORK-N-N*N, INFO ) * SCALEM = WORK(N+N*N+1) NUMRANK = IDNINT(WORK(N+N*N+2)) DO 6970 p = 1, N CALL DCOPY( N, WORK(N+(p-1)*N+1), 1, U(1,p), 1 ) CALL DSCAL( N, SVA(p), WORK(N+(p-1)*N+1), 1 ) 6970 CONTINUE * CALL DTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N, $ ONE, A, LDA, WORK(N+1), N ) DO 6972 p = 1, N CALL DCOPY( N, WORK(N+p), N, V(IWORK(p),1), LDV ) 6972 CONTINUE TEMP1 = DSQRT(DBLE(N))*EPSLN DO 6971 p = 1, N XSC = ONE / DNRM2( N, V(1,p), 1 ) IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) ) $ CALL DSCAL( N, XSC, V(1,p), 1 ) 6971 CONTINUE * * Assemble the left singular vector matrix U (M x N). * IF ( N .LT. M ) THEN CALL DLASET( 'A', M-N, N, ZERO, ZERO, U(N+1,1), LDU ) IF ( N .LT. N1 ) THEN CALL DLASET( 'A',N, N1-N, ZERO, ZERO, U(1,N+1),LDU ) CALL DLASET( 'A',M-N,N1-N, ZERO, ONE,U(N+1,N+1),LDU ) END IF END IF CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U, $ LDU, WORK(N+1), LWORK-N, IERR ) TEMP1 = DSQRT(DBLE(M))*EPSLN DO 6973 p = 1, N1 XSC = ONE / DNRM2( M, U(1,p), 1 ) IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) ) $ CALL DSCAL( M, XSC, U(1,p), 1 ) 6973 CONTINUE * IF ( ROWPIV ) $ CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 ) * END IF * * end of the >> almost orthogonal case << in the full SVD * ELSE * * This branch deploys a preconditioned Jacobi SVD with explicitly * accumulated rotations. It is included as optional, mainly for * experimental purposes. It does perfom well, and can also be used. * In this implementation, this branch will be automatically activated * if the condition number sigma_max(A) / sigma_min(A) is predicted * to be greater than the overflow threshold. This is because the * a posteriori computation of the singular vectors assumes robust * implementation of BLAS and some LAPACK procedures, capable of working * in presence of extreme values. Since that is not always the case, ... * DO 7968 p = 1, NR CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 ) 7968 CONTINUE * IF ( L2PERT ) THEN XSC = DSQRT(SMALL/EPSLN) DO 5969 q = 1, NR TEMP1 = XSC*DABS( V(q,q) ) DO 5968 p = 1, N IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 ) $ .OR. ( p .LT. q ) ) $ V(p,q) = DSIGN( TEMP1, V(p,q) ) IF ( p .LT. q ) V(p,q) = - V(p,q) 5968 CONTINUE 5969 CONTINUE ELSE CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV ) END IF CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1), $ LWORK-2*N, IERR ) CALL DLACPY( 'L', N, NR, V, LDV, WORK(2*N+1), N ) * DO 7969 p = 1, NR CALL DCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 ) 7969 CONTINUE IF ( L2PERT ) THEN XSC = DSQRT(SMALL/EPSLN) DO 9970 q = 2, NR DO 9971 p = 1, q - 1 TEMP1 = XSC * DMIN1(DABS(U(p,p)),DABS(U(q,q))) U(p,q) = - DSIGN( TEMP1, U(q,p) ) 9971 CONTINUE 9970 CONTINUE ELSE CALL DLASET('U', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU ) END IF CALL DGESVJ( 'G', 'U', 'V', NR, NR, U, LDU, SVA, $ N, V, LDV, WORK(2*N+N*NR+1), LWORK-2*N-N*NR, INFO ) SCALEM = WORK(2*N+N*NR+1) NUMRANK = IDNINT(WORK(2*N+N*NR+2)) IF ( NR .LT. N ) THEN CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV ) CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV ) CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV ) END IF CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1), $ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR ) * * Permute the rows of V using the (column) permutation from the * first QRF. Also, scale the columns to make them unit in * Euclidean norm. This applies to all cases. * TEMP1 = DSQRT(DBLE(N)) * EPSLN DO 7972 q = 1, N DO 8972 p = 1, N WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q) 8972 CONTINUE DO 8973 p = 1, N V(p,q) = WORK(2*N+N*NR+NR+p) 8973 CONTINUE XSC = ONE / DNRM2( N, V(1,q), 1 ) IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) ) $ CALL DSCAL( N, XSC, V(1,q), 1 ) 7972 CONTINUE * * At this moment, V contains the right singular vectors of A. * Next, assemble the left singular vector matrix U (M x N). * IF ( NR .LT. M ) THEN CALL DLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU ) IF ( NR .LT. N1 ) THEN CALL DLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1),LDU ) CALL DLASET( 'A',M-NR,N1-NR, ZERO, ONE,U(NR+1,NR+1),LDU ) END IF END IF * CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U, $ LDU, WORK(N+1), LWORK-N, IERR ) * IF ( ROWPIV ) $ CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 ) * * END IF IF ( TRANSP ) THEN * .. swap U and V because the procedure worked on A^t DO 6974 p = 1, N CALL DSWAP( N, U(1,p), 1, V(1,p), 1 ) 6974 CONTINUE END IF * END IF * end of the full SVD * * Undo scaling, if necessary (and possible) * IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN CALL DLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR ) USCAL1 = ONE USCAL2 = ONE END IF * IF ( NR .LT. N ) THEN DO 3004 p = NR+1, N SVA(p) = ZERO 3004 CONTINUE END IF * WORK(1) = USCAL2 * SCALEM WORK(2) = USCAL1 IF ( ERREST ) WORK(3) = SCONDA IF ( LSVEC .AND. RSVEC ) THEN WORK(4) = CONDR1 WORK(5) = CONDR2 END IF IF ( L2TRAN ) THEN WORK(6) = ENTRA WORK(7) = ENTRAT END IF * IWORK(1) = NR IWORK(2) = NUMRANK IWORK(3) = WARNING * RETURN * .. * .. END OF DGEJSV * .. END * |