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SUBROUTINE SGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
$ LDV, WORK, LWORK, 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. * IMPLICIT NONE * .. * .. Scalar Arguments .. INTEGER INFO, LDA, LDV, LWORK, M, MV, N CHARACTER*1 JOBA, JOBU, JOBV * .. * .. Array Arguments .. REAL A( LDA, * ), SVA( N ), V( LDV, * ), $ WORK( LWORK ) * .. * * Purpose * ======= * * SGESVJ computes the singular value decomposition (SVD) of a real * M-by-N matrix A, where M >= N. The SVD of A is written as * [++] [xx] [x0] [xx] * A = U * SIGMA * V^t, [++] = [xx] * [ox] * [xx] * [++] [xx] * where SIGMA is an N-by-N diagonal matrix, U is an M-by-N 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. * * Further Details * ~~~~~~~~~~~~~~~ * The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane * rotations. The rotations are implemented as fast scaled rotations of * Anda and Park [1]. In the case of underflow of the Jacobi angle, a * modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses * column interchanges of de Rijk [2]. The relative accuracy of the computed * singular values and the accuracy of the computed singular vectors (in * angle metric) is as guaranteed by the theory of Demmel and Veselic [3]. * The condition number that determines the accuracy in the full rank case * is essentially min_{D=diag} kappa(A*D), where kappa(.) is the * spectral condition number. The best performance of this Jacobi SVD * procedure is achieved if used in an accelerated version of Drmac and * Veselic [5,6], and it is the kernel routine in the SIGMA library [7]. * Some tunning parameters (marked with [TP]) are available for the * implementer. * The computational range for the nonzero singular values is the machine * number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even * denormalized singular values can be computed with the corresponding * gradual loss of accurate digits. * * Contributors * ~~~~~~~~~~~~ * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany) * * References * ~~~~~~~~~~ * [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling. * SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174. * [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the * singular value decomposition on a vector computer. * SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371. * [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR. * [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular * value computation in floating point arithmetic. * SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222. * [5] 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. * [6] 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. * [7] 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 test examples and comments to * drmac@math.hr. Thank you. * * Arguments * ========= * * JOBA (input) CHARACTER* 1 * Specifies the structure of A. * = 'L': The input matrix A is lower triangular; * = 'U': The input matrix A is upper triangular; * = 'G': The input matrix A is general M-by-N matrix, M >= N. * * JOBU (input) CHARACTER*1 * Specifies whether to compute the left singular vectors * (columns of U): * = 'U': The left singular vectors corresponding to the nonzero * singular values are computed and returned in the leading * columns of A. See more details in the description of A. * The default numerical orthogonality threshold is set to * approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E'). * = 'C': Analogous to JOBU='U', except that user can control the * level of numerical orthogonality of the computed left * singular vectors. TOL can be set to TOL = CTOL*EPS, where * CTOL is given on input in the array WORK. * No CTOL smaller than ONE is allowed. CTOL greater * than 1 / EPS is meaningless. The option 'C' * can be used if M*EPS is satisfactory orthogonality * of the computed left singular vectors, so CTOL=M could * save few sweeps of Jacobi rotations. * See the descriptions of A and WORK(1). * = 'N': The matrix U is not computed. However, see the * description of A. * * JOBV (input) CHARACTER*1 * Specifies whether to compute the right singular vectors, that * is, the matrix V: * = 'V' : the matrix V is computed and returned in the array V * = 'A' : the Jacobi rotations are applied to the MV-by-N * array V. In other words, the right singular vector * matrix V is not computed explicitly; instead it is * applied to an MV-by-N matrix initially stored in the * first MV rows of V. * = 'N' : the matrix V is not computed and the array V is not * referenced * * M (input) INTEGER * The number of rows of the input matrix A. 1/SLAMCH('E') > M >= 0. * * N (input) INTEGER * The number of columns of the input matrix A. * M >= N >= 0. * * A (input/output) REAL array, dimension (LDA,N) * On entry, the M-by-N matrix A. * On exit, * If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C': * If INFO .EQ. 0 : * RANKA orthonormal columns of U are returned in the * leading RANKA columns of the array A. Here RANKA <= N * is the number of computed singular values of A that are * above the underflow threshold SLAMCH('S'). The singular * vectors corresponding to underflowed or zero singular * values are not computed. The value of RANKA is returned * in the array WORK as RANKA=NINT(WORK(2)). Also see the * descriptions of SVA and WORK. The computed columns of U * are mutually numerically orthogonal up to approximately * TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'), * see the description of JOBU. * If INFO .GT. 0, * the procedure SGESVJ did not converge in the given number * of iterations (sweeps). In that case, the computed * columns of U may not be orthogonal up to TOL. The output * U (stored in A), SIGMA (given by the computed singular * values in SVA(1:N)) and V is still a decomposition of the * input matrix A in the sense that the residual * ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small. * If JOBU .EQ. 'N': * If INFO .EQ. 0 : * Note that the left singular vectors are 'for free' in the * one-sided Jacobi SVD algorithm. However, if only the * singular values are needed, the level of numerical * orthogonality of U is not an issue and iterations are * stopped when the columns of the iterated matrix are * numerically orthogonal up to approximately M*EPS. Thus, * on exit, A contains the columns of U scaled with the * corresponding singular values. * If INFO .GT. 0 : * the procedure SGESVJ did not converge in the given number * of iterations (sweeps). * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * SVA (workspace/output) REAL array, dimension (N) * On exit, * If INFO .EQ. 0 : * depending on the value SCALE = WORK(1), we have: * If SCALE .EQ. ONE: * SVA(1:N) contains the computed singular values of A. * During the computation SVA contains the Euclidean column * norms of the iterated matrices in the array A. * If SCALE .NE. ONE: * The singular values of A are SCALE*SVA(1:N), and this * factored representation is due to the fact that some of the * singular values of A might underflow or overflow. * * If INFO .GT. 0 : * the procedure SGESVJ did not converge in the given number of * iterations (sweeps) and SCALE*SVA(1:N) may not be accurate. * * MV (input) INTEGER * If JOBV .EQ. 'A', then the product of Jacobi rotations in SGESVJ * is applied to the first MV rows of V. See the description of JOBV. * * V (input/output) REAL array, dimension (LDV,N) * If JOBV = 'V', then V contains on exit the N-by-N matrix of * the right singular vectors; * If JOBV = 'A', then V contains the product of the computed right * singular vector matrix and the initial matrix in * the array V. * If JOBV = 'N', then V is not referenced. * * LDV (input) INTEGER * The leading dimension of the array V, LDV .GE. 1. * If JOBV .EQ. 'V', then LDV .GE. max(1,N). * If JOBV .EQ. 'A', then LDV .GE. max(1,MV) . * * WORK (input/workspace/output) REAL array, dimension max(4,M+N). * On entry, * If JOBU .EQ. 'C' : * WORK(1) = CTOL, where CTOL defines the threshold for convergence. * The process stops if all columns of A are mutually * orthogonal up to CTOL*EPS, EPS=SLAMCH('E'). * It is required that CTOL >= ONE, i.e. it is not * allowed to force the routine to obtain orthogonality * below EPSILON. * On exit, * WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N) * are the computed singular vcalues of A. * (See description of SVA().) * WORK(2) = NINT(WORK(2)) is the number of the computed nonzero * singular values. * WORK(3) = NINT(WORK(3)) is the number of the computed singular * values that are larger than the underflow threshold. * WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi * rotations needed for numerical convergence. * WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep. * This is useful information in cases when SGESVJ did * not converge, as it can be used to estimate whether * the output is stil useful and for post festum analysis. * WORK(6) = the largest absolute value over all sines of the * Jacobi rotation angles in the last sweep. It can be * useful for a post festum analysis. * * LWORK (input) INTEGER * length of WORK, WORK >= MAX(6,M+N) * * INFO (output) INTEGER * = 0 : successful exit. * < 0 : if INFO = -i, then the i-th argument had an illegal value * > 0 : SGESVJ did not converge in the maximal allowed number (30) * of sweeps. The output may still be useful. See the * description of WORK. * * ===================================================================== * * .. Local Parameters .. REAL ZERO, HALF, ONE, TWO PARAMETER ( ZERO = 0.0E0, HALF = 0.5E0, ONE = 1.0E0, $ TWO = 2.0E0 ) INTEGER NSWEEP PARAMETER ( NSWEEP = 30 ) * .. * .. Local Scalars .. REAL AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG, $ BIGTHETA, CS, CTOL, EPSLN, LARGE, MXAAPQ, $ MXSINJ, ROOTBIG, ROOTEPS, ROOTSFMIN, ROOTTOL, $ SKL, SFMIN, SMALL, SN, T, TEMP1, THETA, $ THSIGN, TOL INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1, $ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, N2, N34, $ N4, NBL, NOTROT, p, PSKIPPED, q, ROWSKIP, $ SWBAND LOGICAL APPLV, GOSCALE, LOWER, LSVEC, NOSCALE, ROTOK, $ RSVEC, UCTOL, UPPER * .. * .. Local Arrays .. REAL FASTR( 5 ) * .. * .. Intrinsic Functions .. INTRINSIC ABS, AMAX1, AMIN1, FLOAT, MIN0, SIGN, SQRT * .. * .. External Functions .. * .. * from BLAS REAL SDOT, SNRM2 EXTERNAL SDOT, SNRM2 INTEGER ISAMAX EXTERNAL ISAMAX * from LAPACK REAL SLAMCH EXTERNAL SLAMCH LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. * .. * from BLAS EXTERNAL SAXPY, SCOPY, SROTM, SSCAL, SSWAP * from LAPACK EXTERNAL SLASCL, SLASET, SLASSQ, XERBLA * EXTERNAL SGSVJ0, SGSVJ1 * .. * .. Executable Statements .. * * Test the input arguments * LSVEC = LSAME( JOBU, 'U' ) UCTOL = LSAME( JOBU, 'C' ) RSVEC = LSAME( JOBV, 'V' ) APPLV = LSAME( JOBV, 'A' ) UPPER = LSAME( JOBA, 'U' ) LOWER = LSAME( JOBA, 'L' ) * IF( .NOT.( UPPER .OR. LOWER .OR. LSAME( JOBA, 'G' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( LSVEC .OR. UCTOL .OR. LSAME( JOBU, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN INFO = -5 ELSE IF( LDA.LT.M ) THEN INFO = -7 ELSE IF( MV.LT.0 ) THEN INFO = -9 ELSE IF( ( RSVEC .AND. ( LDV.LT.N ) ) .OR. $ ( APPLV .AND. ( LDV.LT.MV ) ) ) THEN INFO = -11 ELSE IF( UCTOL .AND. ( WORK( 1 ).LE.ONE ) ) THEN INFO = -12 ELSE IF( LWORK.LT.MAX0( M+N, 6 ) ) THEN INFO = -13 ELSE INFO = 0 END IF * * #:( IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGESVJ', -INFO ) RETURN END IF * * #:) Quick return for void matrix * IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )RETURN * * Set numerical parameters * The stopping criterion for Jacobi rotations is * * max_{i<>j}|A(:,i)^T * A(:,j)|/(||A(:,i)||*||A(:,j)||) < CTOL*EPS * * where EPS is the round-off and CTOL is defined as follows: * IF( UCTOL ) THEN * ... user controlled CTOL = WORK( 1 ) ELSE * ... default IF( LSVEC .OR. RSVEC .OR. APPLV ) THEN CTOL = SQRT( FLOAT( M ) ) ELSE CTOL = FLOAT( M ) END IF END IF * ... and the machine dependent parameters are *[!] (Make sure that SLAMCH() works properly on the target machine.) * EPSLN = SLAMCH( 'Epsilon' ) ROOTEPS = SQRT( EPSLN ) SFMIN = SLAMCH( 'SafeMinimum' ) ROOTSFMIN = SQRT( SFMIN ) SMALL = SFMIN / EPSLN BIG = SLAMCH( 'Overflow' ) * BIG = ONE / SFMIN ROOTBIG = ONE / ROOTSFMIN LARGE = BIG / SQRT( FLOAT( M*N ) ) BIGTHETA = ONE / ROOTEPS * TOL = CTOL*EPSLN ROOTTOL = SQRT( TOL ) * IF( FLOAT( M )*EPSLN.GE.ONE ) THEN INFO = -4 CALL XERBLA( 'SGESVJ', -INFO ) RETURN END IF * * Initialize the right singular vector matrix. * IF( RSVEC ) THEN MVL = N CALL SLASET( 'A', MVL, N, ZERO, ONE, V, LDV ) ELSE IF( APPLV ) THEN MVL = MV END IF RSVEC = RSVEC .OR. APPLV * * Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N ) *(!) If necessary, scale A to protect the largest singular value * from overflow. It is possible that saving the largest singular * value destroys the information about the small ones. * This initial scaling is almost minimal in the sense that the * goal is to make sure that no column norm overflows, and that * SQRT(N)*max_i SVA(i) does not overflow. If INFinite entries * in A are detected, the procedure returns with INFO=-6. * SKL = ONE / SQRT( FLOAT( M )*FLOAT( N ) ) NOSCALE = .TRUE. GOSCALE = .TRUE. * IF( LOWER ) THEN * the input matrix is M-by-N lower triangular (trapezoidal) DO 1874 p = 1, N AAPP = ZERO AAQQ = ONE CALL SLASSQ( M-p+1, A( p, p ), 1, AAPP, AAQQ ) IF( AAPP.GT.BIG ) THEN INFO = -6 CALL XERBLA( 'SGESVJ', -INFO ) RETURN END IF AAQQ = SQRT( AAQQ ) IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN SVA( p ) = AAPP*AAQQ ELSE NOSCALE = .FALSE. SVA( p ) = AAPP*( AAQQ*SKL ) IF( GOSCALE ) THEN GOSCALE = .FALSE. DO 1873 q = 1, p - 1 SVA( q ) = SVA( q )*SKL 1873 CONTINUE END IF END IF 1874 CONTINUE ELSE IF( UPPER ) THEN * the input matrix is M-by-N upper triangular (trapezoidal) DO 2874 p = 1, N AAPP = ZERO AAQQ = ONE CALL SLASSQ( p, A( 1, p ), 1, AAPP, AAQQ ) IF( AAPP.GT.BIG ) THEN INFO = -6 CALL XERBLA( 'SGESVJ', -INFO ) RETURN END IF AAQQ = SQRT( AAQQ ) IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN SVA( p ) = AAPP*AAQQ ELSE NOSCALE = .FALSE. SVA( p ) = AAPP*( AAQQ*SKL ) IF( GOSCALE ) THEN GOSCALE = .FALSE. DO 2873 q = 1, p - 1 SVA( q ) = SVA( q )*SKL 2873 CONTINUE END IF END IF 2874 CONTINUE ELSE * the input matrix is M-by-N general dense DO 3874 p = 1, N AAPP = ZERO AAQQ = ONE CALL SLASSQ( M, A( 1, p ), 1, AAPP, AAQQ ) IF( AAPP.GT.BIG ) THEN INFO = -6 CALL XERBLA( 'SGESVJ', -INFO ) RETURN END IF AAQQ = SQRT( AAQQ ) IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN SVA( p ) = AAPP*AAQQ ELSE NOSCALE = .FALSE. SVA( p ) = AAPP*( AAQQ*SKL ) IF( GOSCALE ) THEN GOSCALE = .FALSE. DO 3873 q = 1, p - 1 SVA( q ) = SVA( q )*SKL 3873 CONTINUE END IF END IF 3874 CONTINUE END IF * IF( NOSCALE )SKL = ONE * * Move the smaller part of the spectrum from the underflow threshold *(!) Start by determining the position of the nonzero entries of the * array SVA() relative to ( SFMIN, BIG ). * AAPP = ZERO AAQQ = BIG DO 4781 p = 1, N IF( SVA( p ).NE.ZERO )AAQQ = AMIN1( AAQQ, SVA( p ) ) AAPP = AMAX1( AAPP, SVA( p ) ) 4781 CONTINUE * * #:) Quick return for zero matrix * IF( AAPP.EQ.ZERO ) THEN IF( LSVEC )CALL SLASET( 'G', M, N, ZERO, ONE, A, LDA ) WORK( 1 ) = ONE WORK( 2 ) = ZERO WORK( 3 ) = ZERO WORK( 4 ) = ZERO WORK( 5 ) = ZERO WORK( 6 ) = ZERO RETURN END IF * * #:) Quick return for one-column matrix * IF( N.EQ.1 ) THEN IF( LSVEC )CALL SLASCL( 'G', 0, 0, SVA( 1 ), SKL, M, 1, $ A( 1, 1 ), LDA, IERR ) WORK( 1 ) = ONE / SKL IF( SVA( 1 ).GE.SFMIN ) THEN WORK( 2 ) = ONE ELSE WORK( 2 ) = ZERO END IF WORK( 3 ) = ZERO WORK( 4 ) = ZERO WORK( 5 ) = ZERO WORK( 6 ) = ZERO RETURN END IF * * Protect small singular values from underflow, and try to * avoid underflows/overflows in computing Jacobi rotations. * SN = SQRT( SFMIN / EPSLN ) TEMP1 = SQRT( BIG / FLOAT( N ) ) IF( ( AAPP.LE.SN ) .OR. ( AAQQ.GE.TEMP1 ) .OR. $ ( ( SN.LE.AAQQ ) .AND. ( AAPP.LE.TEMP1 ) ) ) THEN TEMP1 = AMIN1( BIG, TEMP1 / AAPP ) * AAQQ = AAQQ*TEMP1 * AAPP = AAPP*TEMP1 ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.LE.TEMP1 ) ) THEN TEMP1 = AMIN1( SN / AAQQ, BIG / ( AAPP*SQRT( FLOAT( N ) ) ) ) * AAQQ = AAQQ*TEMP1 * AAPP = AAPP*TEMP1 ELSE IF( ( AAQQ.GE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN TEMP1 = AMAX1( SN / AAQQ, TEMP1 / AAPP ) * AAQQ = AAQQ*TEMP1 * AAPP = AAPP*TEMP1 ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN TEMP1 = AMIN1( SN / AAQQ, BIG / ( SQRT( FLOAT( N ) )*AAPP ) ) * AAQQ = AAQQ*TEMP1 * AAPP = AAPP*TEMP1 ELSE TEMP1 = ONE END IF * * Scale, if necessary * IF( TEMP1.NE.ONE ) THEN CALL SLASCL( 'G', 0, 0, ONE, TEMP1, N, 1, SVA, N, IERR ) END IF SKL = TEMP1*SKL IF( SKL.NE.ONE ) THEN CALL SLASCL( JOBA, 0, 0, ONE, SKL, M, N, A, LDA, IERR ) SKL = ONE / SKL END IF * * Row-cyclic Jacobi SVD algorithm with column pivoting * EMPTSW = ( N*( N-1 ) ) / 2 NOTROT = 0 FASTR( 1 ) = ZERO * * A is represented in factored form A = A * diag(WORK), where diag(WORK) * is initialized to identity. WORK is updated during fast scaled * rotations. * DO 1868 q = 1, N WORK( q ) = ONE 1868 CONTINUE * * SWBAND = 3 *[TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective * if SGESVJ is used as a computational routine in the preconditioned * Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure * works on pivots inside a band-like region around the diagonal. * The boundaries are determined dynamically, based on the number of * pivots above a threshold. * KBL = MIN0( 8, N ) *[TP] KBL is a tuning parameter that defines the tile size in the * tiling of the p-q loops of pivot pairs. In general, an optimal * value of KBL depends on the matrix dimensions and on the * parameters of the computer's memory. * NBL = N / KBL IF( ( NBL*KBL ).NE.N )NBL = NBL + 1 * BLSKIP = KBL**2 *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL. * ROWSKIP = MIN0( 5, KBL ) *[TP] ROWSKIP is a tuning parameter. * LKAHEAD = 1 *[TP] LKAHEAD is a tuning parameter. * * Quasi block transformations, using the lower (upper) triangular * structure of the input matrix. The quasi-block-cycling usually * invokes cubic convergence. Big part of this cycle is done inside * canonical subspaces of dimensions less than M. * IF( ( LOWER .OR. UPPER ) .AND. ( N.GT.MAX0( 64, 4*KBL ) ) ) THEN *[TP] The number of partition levels and the actual partition are * tuning parameters. N4 = N / 4 N2 = N / 2 N34 = 3*N4 IF( APPLV ) THEN q = 0 ELSE q = 1 END IF * IF( LOWER ) THEN * * This works very well on lower triangular matrices, in particular * in the framework of the preconditioned Jacobi SVD (xGEJSV). * The idea is simple: * [+ 0 0 0] Note that Jacobi transformations of [0 0] * [+ + 0 0] [0 0] * [+ + x 0] actually work on [x 0] [x 0] * [+ + x x] [x x]. [x x] * CALL SGSVJ0( JOBV, M-N34, N-N34, A( N34+1, N34+1 ), LDA, $ WORK( N34+1 ), SVA( N34+1 ), MVL, $ V( N34*q+1, N34+1 ), LDV, EPSLN, SFMIN, TOL, $ 2, WORK( N+1 ), LWORK-N, IERR ) * CALL SGSVJ0( JOBV, M-N2, N34-N2, A( N2+1, N2+1 ), LDA, $ WORK( N2+1 ), SVA( N2+1 ), MVL, $ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 2, $ WORK( N+1 ), LWORK-N, IERR ) * CALL SGSVJ1( JOBV, M-N2, N-N2, N4, A( N2+1, N2+1 ), LDA, $ WORK( N2+1 ), SVA( N2+1 ), MVL, $ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1, $ WORK( N+1 ), LWORK-N, IERR ) * CALL SGSVJ0( JOBV, M-N4, N2-N4, A( N4+1, N4+1 ), LDA, $ WORK( N4+1 ), SVA( N4+1 ), MVL, $ V( N4*q+1, N4+1 ), LDV, EPSLN, SFMIN, TOL, 1, $ WORK( N+1 ), LWORK-N, IERR ) * CALL SGSVJ0( JOBV, M, N4, A, LDA, WORK, SVA, MVL, V, LDV, $ EPSLN, SFMIN, TOL, 1, WORK( N+1 ), LWORK-N, $ IERR ) * CALL SGSVJ1( JOBV, M, N2, N4, A, LDA, WORK, SVA, MVL, V, $ LDV, EPSLN, SFMIN, TOL, 1, WORK( N+1 ), $ LWORK-N, IERR ) * * ELSE IF( UPPER ) THEN * * CALL SGSVJ0( JOBV, N4, N4, A, LDA, WORK, SVA, MVL, V, LDV, $ EPSLN, SFMIN, TOL, 2, WORK( N+1 ), LWORK-N, $ IERR ) * CALL SGSVJ0( JOBV, N2, N4, A( 1, N4+1 ), LDA, WORK( N4+1 ), $ SVA( N4+1 ), MVL, V( N4*q+1, N4+1 ), LDV, $ EPSLN, SFMIN, TOL, 1, WORK( N+1 ), LWORK-N, $ IERR ) * CALL SGSVJ1( JOBV, N2, N2, N4, A, LDA, WORK, SVA, MVL, V, $ LDV, EPSLN, SFMIN, TOL, 1, WORK( N+1 ), $ LWORK-N, IERR ) * CALL SGSVJ0( JOBV, N2+N4, N4, A( 1, N2+1 ), LDA, $ WORK( N2+1 ), SVA( N2+1 ), MVL, $ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1, $ WORK( N+1 ), LWORK-N, IERR ) END IF * END IF * * .. Row-cyclic pivot strategy with de Rijk's pivoting .. * DO 1993 i = 1, NSWEEP * * .. go go go ... * MXAAPQ = ZERO MXSINJ = ZERO ISWROT = 0 * NOTROT = 0 PSKIPPED = 0 * * Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs * 1 <= p < q <= N. This is the first step toward a blocked implementation * of the rotations. New implementation, based on block transformations, * is under development. * DO 2000 ibr = 1, NBL * igl = ( ibr-1 )*KBL + 1 * DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr ) * igl = igl + ir1*KBL * DO 2001 p = igl, MIN0( igl+KBL-1, N-1 ) * * .. de Rijk's pivoting * q = ISAMAX( N-p+1, SVA( p ), 1 ) + p - 1 IF( p.NE.q ) THEN CALL SSWAP( M, A( 1, p ), 1, A( 1, q ), 1 ) IF( RSVEC )CALL SSWAP( MVL, V( 1, p ), 1, $ V( 1, q ), 1 ) TEMP1 = SVA( p ) SVA( p ) = SVA( q ) SVA( q ) = TEMP1 TEMP1 = WORK( p ) WORK( p ) = WORK( q ) WORK( q ) = TEMP1 END IF * IF( ir1.EQ.0 ) THEN * * Column norms are periodically updated by explicit * norm computation. * Caveat: * Unfortunately, some BLAS implementations compute SNRM2(M,A(1,p),1) * as SQRT(SDOT(M,A(1,p),1,A(1,p),1)), which may cause the result to * overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and to * underflow for ||A(:,p)||_2 < SQRT(underflow_threshold). * Hence, SNRM2 cannot be trusted, not even in the case when * the true norm is far from the under(over)flow boundaries. * If properly implemented SNRM2 is available, the IF-THEN-ELSE * below should read "AAPP = SNRM2( M, A(1,p), 1 ) * WORK(p)". * IF( ( SVA( p ).LT.ROOTBIG ) .AND. $ ( SVA( p ).GT.ROOTSFMIN ) ) THEN SVA( p ) = SNRM2( M, A( 1, p ), 1 )*WORK( p ) ELSE TEMP1 = ZERO AAPP = ONE CALL SLASSQ( M, A( 1, p ), 1, TEMP1, AAPP ) SVA( p ) = TEMP1*SQRT( AAPP )*WORK( p ) END IF AAPP = SVA( p ) ELSE AAPP = SVA( p ) END IF * IF( AAPP.GT.ZERO ) THEN * PSKIPPED = 0 * DO 2002 q = p + 1, MIN0( igl+KBL-1, N ) * AAQQ = SVA( q ) * IF( AAQQ.GT.ZERO ) THEN * AAPP0 = AAPP IF( AAQQ.GE.ONE ) THEN ROTOK = ( SMALL*AAPP ).LE.AAQQ IF( AAPP.LT.( BIG / AAQQ ) ) THEN AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1, $ q ), 1 )*WORK( p )*WORK( q ) / $ AAQQ ) / AAPP ELSE CALL SCOPY( M, A( 1, p ), 1, $ WORK( N+1 ), 1 ) CALL SLASCL( 'G', 0, 0, AAPP, $ WORK( p ), M, 1, $ WORK( N+1 ), LDA, IERR ) AAPQ = SDOT( M, WORK( N+1 ), 1, $ A( 1, q ), 1 )*WORK( q ) / AAQQ END IF ELSE ROTOK = AAPP.LE.( AAQQ / SMALL ) IF( AAPP.GT.( SMALL / AAQQ ) ) THEN AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1, $ q ), 1 )*WORK( p )*WORK( q ) / $ AAQQ ) / AAPP ELSE CALL SCOPY( M, A( 1, q ), 1, $ WORK( N+1 ), 1 ) CALL SLASCL( 'G', 0, 0, AAQQ, $ WORK( q ), M, 1, $ WORK( N+1 ), LDA, IERR ) AAPQ = SDOT( M, WORK( N+1 ), 1, $ A( 1, p ), 1 )*WORK( p ) / AAPP END IF END IF * MXAAPQ = AMAX1( MXAAPQ, ABS( AAPQ ) ) * * TO rotate or NOT to rotate, THAT is the question ... * IF( ABS( AAPQ ).GT.TOL ) THEN * * .. rotate *[RTD] ROTATED = ROTATED + ONE * IF( ir1.EQ.0 ) THEN NOTROT = 0 PSKIPPED = 0 ISWROT = ISWROT + 1 END IF * IF( ROTOK ) THEN * AQOAP = AAQQ / AAPP APOAQ = AAPP / AAQQ THETA = -HALF*ABS( AQOAP-APOAQ ) / AAPQ * IF( ABS( THETA ).GT.BIGTHETA ) THEN * T = HALF / THETA FASTR( 3 ) = T*WORK( p ) / WORK( q ) FASTR( 4 ) = -T*WORK( q ) / $ WORK( p ) CALL SROTM( M, A( 1, p ), 1, $ A( 1, q ), 1, FASTR ) IF( RSVEC )CALL SROTM( MVL, $ V( 1, p ), 1, $ V( 1, q ), 1, $ FASTR ) SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, $ ONE+T*APOAQ*AAPQ ) ) AAPP = AAPP*SQRT( AMAX1( ZERO, $ ONE-T*AQOAP*AAPQ ) ) MXSINJ = AMAX1( MXSINJ, ABS( T ) ) * ELSE * * .. choose correct signum for THETA and rotate * THSIGN = -SIGN( ONE, AAPQ ) T = ONE / ( THETA+THSIGN* $ SQRT( ONE+THETA*THETA ) ) CS = SQRT( ONE / ( ONE+T*T ) ) SN = T*CS * MXSINJ = AMAX1( MXSINJ, ABS( SN ) ) SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, $ ONE+T*APOAQ*AAPQ ) ) AAPP = AAPP*SQRT( AMAX1( ZERO, $ ONE-T*AQOAP*AAPQ ) ) * APOAQ = WORK( p ) / WORK( q ) AQOAP = WORK( q ) / WORK( p ) IF( WORK( p ).GE.ONE ) THEN IF( WORK( q ).GE.ONE ) THEN FASTR( 3 ) = T*APOAQ FASTR( 4 ) = -T*AQOAP WORK( p ) = WORK( p )*CS WORK( q ) = WORK( q )*CS CALL SROTM( M, A( 1, p ), 1, $ A( 1, q ), 1, $ FASTR ) IF( RSVEC )CALL SROTM( MVL, $ V( 1, p ), 1, V( 1, q ), $ 1, FASTR ) ELSE CALL SAXPY( M, -T*AQOAP, $ A( 1, q ), 1, $ A( 1, p ), 1 ) CALL SAXPY( M, CS*SN*APOAQ, $ A( 1, p ), 1, $ A( 1, q ), 1 ) WORK( p ) = WORK( p )*CS WORK( q ) = WORK( q ) / CS IF( RSVEC ) THEN CALL SAXPY( MVL, -T*AQOAP, $ V( 1, q ), 1, $ V( 1, p ), 1 ) CALL SAXPY( MVL, $ CS*SN*APOAQ, $ V( 1, p ), 1, $ V( 1, q ), 1 ) END IF END IF ELSE IF( WORK( q ).GE.ONE ) THEN CALL SAXPY( M, T*APOAQ, $ A( 1, p ), 1, $ A( 1, q ), 1 ) CALL SAXPY( M, -CS*SN*AQOAP, $ A( 1, q ), 1, $ A( 1, p ), 1 ) WORK( p ) = WORK( p ) / CS WORK( q ) = WORK( q )*CS IF( RSVEC ) THEN CALL SAXPY( MVL, T*APOAQ, $ V( 1, p ), 1, $ V( 1, q ), 1 ) CALL SAXPY( MVL, $ -CS*SN*AQOAP, $ V( 1, q ), 1, $ V( 1, p ), 1 ) END IF ELSE IF( WORK( p ).GE.WORK( q ) ) $ THEN CALL SAXPY( M, -T*AQOAP, $ A( 1, q ), 1, $ A( 1, p ), 1 ) CALL SAXPY( M, CS*SN*APOAQ, $ A( 1, p ), 1, $ A( 1, q ), 1 ) WORK( p ) = WORK( p )*CS WORK( q ) = WORK( q ) / CS IF( RSVEC ) THEN CALL SAXPY( MVL, $ -T*AQOAP, $ V( 1, q ), 1, $ V( 1, p ), 1 ) CALL SAXPY( MVL, $ CS*SN*APOAQ, $ V( 1, p ), 1, $ V( 1, q ), 1 ) END IF ELSE CALL SAXPY( M, T*APOAQ, $ A( 1, p ), 1, $ A( 1, q ), 1 ) CALL SAXPY( M, $ -CS*SN*AQOAP, $ A( 1, q ), 1, $ A( 1, p ), 1 ) WORK( p ) = WORK( p ) / CS WORK( q ) = WORK( q )*CS IF( RSVEC ) THEN CALL SAXPY( MVL, $ T*APOAQ, V( 1, p ), $ 1, V( 1, q ), 1 ) CALL SAXPY( MVL, $ -CS*SN*AQOAP, $ V( 1, q ), 1, $ V( 1, p ), 1 ) END IF END IF END IF END IF END IF * ELSE * .. have to use modified Gram-Schmidt like transformation CALL SCOPY( M, A( 1, p ), 1, $ WORK( N+1 ), 1 ) CALL SLASCL( 'G', 0, 0, AAPP, ONE, M, $ 1, WORK( N+1 ), LDA, $ IERR ) CALL SLASCL( 'G', 0, 0, AAQQ, ONE, M, $ 1, A( 1, q ), LDA, IERR ) TEMP1 = -AAPQ*WORK( p ) / WORK( q ) CALL SAXPY( M, TEMP1, WORK( N+1 ), 1, $ A( 1, q ), 1 ) CALL SLASCL( 'G', 0, 0, ONE, AAQQ, M, $ 1, A( 1, q ), LDA, IERR ) SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, $ ONE-AAPQ*AAPQ ) ) MXSINJ = AMAX1( MXSINJ, SFMIN ) END IF * END IF ROTOK THEN ... ELSE * * In the case of cancellation in updating SVA(q), SVA(p) * recompute SVA(q), SVA(p). * IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS ) $ THEN IF( ( AAQQ.LT.ROOTBIG ) .AND. $ ( AAQQ.GT.ROOTSFMIN ) ) THEN SVA( q ) = SNRM2( M, A( 1, q ), 1 )* $ WORK( q ) ELSE T = ZERO AAQQ = ONE CALL SLASSQ( M, A( 1, q ), 1, T, $ AAQQ ) SVA( q ) = T*SQRT( AAQQ )*WORK( q ) END IF END IF IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN IF( ( AAPP.LT.ROOTBIG ) .AND. $ ( AAPP.GT.ROOTSFMIN ) ) THEN AAPP = SNRM2( M, A( 1, p ), 1 )* $ WORK( p ) ELSE T = ZERO AAPP = ONE CALL SLASSQ( M, A( 1, p ), 1, T, $ AAPP ) AAPP = T*SQRT( AAPP )*WORK( p ) END IF SVA( p ) = AAPP END IF * ELSE * A(:,p) and A(:,q) already numerically orthogonal IF( ir1.EQ.0 )NOTROT = NOTROT + 1 *[RTD] SKIPPED = SKIPPED + 1 PSKIPPED = PSKIPPED + 1 END IF ELSE * A(:,q) is zero column IF( ir1.EQ.0 )NOTROT = NOTROT + 1 PSKIPPED = PSKIPPED + 1 END IF * IF( ( i.LE.SWBAND ) .AND. $ ( PSKIPPED.GT.ROWSKIP ) ) THEN IF( ir1.EQ.0 )AAPP = -AAPP NOTROT = 0 GO TO 2103 END IF * 2002 CONTINUE * END q-LOOP * 2103 CONTINUE * bailed out of q-loop * SVA( p ) = AAPP * ELSE SVA( p ) = AAPP IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) ) $ NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p END IF * 2001 CONTINUE * end of the p-loop * end of doing the block ( ibr, ibr ) 1002 CONTINUE * end of ir1-loop * * ... go to the off diagonal blocks * igl = ( ibr-1 )*KBL + 1 * DO 2010 jbc = ibr + 1, NBL * jgl = ( jbc-1 )*KBL + 1 * * doing the block at ( ibr, jbc ) * IJBLSK = 0 DO 2100 p = igl, MIN0( igl+KBL-1, N ) * AAPP = SVA( p ) IF( AAPP.GT.ZERO ) THEN * PSKIPPED = 0 * DO 2200 q = jgl, MIN0( jgl+KBL-1, N ) * AAQQ = SVA( q ) IF( AAQQ.GT.ZERO ) THEN AAPP0 = AAPP * * .. M x 2 Jacobi SVD .. * * Safe Gram matrix computation * IF( AAQQ.GE.ONE ) THEN IF( AAPP.GE.AAQQ ) THEN ROTOK = ( SMALL*AAPP ).LE.AAQQ ELSE ROTOK = ( SMALL*AAQQ ).LE.AAPP END IF IF( AAPP.LT.( BIG / AAQQ ) ) THEN AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1, $ q ), 1 )*WORK( p )*WORK( q ) / $ AAQQ ) / AAPP ELSE CALL SCOPY( M, A( 1, p ), 1, $ WORK( N+1 ), 1 ) CALL SLASCL( 'G', 0, 0, AAPP, $ WORK( p ), M, 1, $ WORK( N+1 ), LDA, IERR ) AAPQ = SDOT( M, WORK( N+1 ), 1, $ A( 1, q ), 1 )*WORK( q ) / AAQQ END IF ELSE IF( AAPP.GE.AAQQ ) THEN ROTOK = AAPP.LE.( AAQQ / SMALL ) ELSE ROTOK = AAQQ.LE.( AAPP / SMALL ) END IF IF( AAPP.GT.( SMALL / AAQQ ) ) THEN AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1, $ q ), 1 )*WORK( p )*WORK( q ) / $ AAQQ ) / AAPP ELSE CALL SCOPY( M, A( 1, q ), 1, $ WORK( N+1 ), 1 ) CALL SLASCL( 'G', 0, 0, AAQQ, $ WORK( q ), M, 1, $ WORK( N+1 ), LDA, IERR ) AAPQ = SDOT( M, WORK( N+1 ), 1, $ A( 1, p ), 1 )*WORK( p ) / AAPP END IF END IF * MXAAPQ = AMAX1( MXAAPQ, ABS( AAPQ ) ) * * TO rotate or NOT to rotate, THAT is the question ... * IF( ABS( AAPQ ).GT.TOL ) THEN NOTROT = 0 *[RTD] ROTATED = ROTATED + 1 PSKIPPED = 0 ISWROT = ISWROT + 1 * IF( ROTOK ) THEN * AQOAP = AAQQ / AAPP APOAQ = AAPP / AAQQ THETA = -HALF*ABS( AQOAP-APOAQ ) / AAPQ IF( AAQQ.GT.AAPP0 )THETA = -THETA * IF( ABS( THETA ).GT.BIGTHETA ) THEN T = HALF / THETA FASTR( 3 ) = T*WORK( p ) / WORK( q ) FASTR( 4 ) = -T*WORK( q ) / $ WORK( p ) CALL SROTM( M, A( 1, p ), 1, $ A( 1, q ), 1, FASTR ) IF( RSVEC )CALL SROTM( MVL, $ V( 1, p ), 1, $ V( 1, q ), 1, $ FASTR ) SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, $ ONE+T*APOAQ*AAPQ ) ) AAPP = AAPP*SQRT( AMAX1( ZERO, $ ONE-T*AQOAP*AAPQ ) ) MXSINJ = AMAX1( MXSINJ, ABS( T ) ) ELSE * * .. choose correct signum for THETA and rotate * THSIGN = -SIGN( ONE, AAPQ ) IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN T = ONE / ( THETA+THSIGN* $ SQRT( ONE+THETA*THETA ) ) CS = SQRT( ONE / ( ONE+T*T ) ) SN = T*CS MXSINJ = AMAX1( MXSINJ, ABS( SN ) ) SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, $ ONE+T*APOAQ*AAPQ ) ) AAPP = AAPP*SQRT( AMAX1( ZERO, $ ONE-T*AQOAP*AAPQ ) ) * APOAQ = WORK( p ) / WORK( q ) AQOAP = WORK( q ) / WORK( p ) IF( WORK( p ).GE.ONE ) THEN * IF( WORK( q ).GE.ONE ) THEN FASTR( 3 ) = T*APOAQ FASTR( 4 ) = -T*AQOAP WORK( p ) = WORK( p )*CS WORK( q ) = WORK( q )*CS CALL SROTM( M, A( 1, p ), 1, $ A( 1, q ), 1, $ FASTR ) IF( RSVEC )CALL SROTM( MVL, $ V( 1, p ), 1, V( 1, q ), $ 1, FASTR ) ELSE CALL SAXPY( M, -T*AQOAP, $ A( 1, q ), 1, $ A( 1, p ), 1 ) CALL SAXPY( M, CS*SN*APOAQ, $ A( 1, p ), 1, $ A( 1, q ), 1 ) IF( RSVEC ) THEN CALL SAXPY( MVL, -T*AQOAP, $ V( 1, q ), 1, $ V( 1, p ), 1 ) CALL SAXPY( MVL, $ CS*SN*APOAQ, $ V( 1, p ), 1, $ V( 1, q ), 1 ) END IF WORK( p ) = WORK( p )*CS WORK( q ) = WORK( q ) / CS END IF ELSE IF( WORK( q ).GE.ONE ) THEN CALL SAXPY( M, T*APOAQ, $ A( 1, p ), 1, $ A( 1, q ), 1 ) CALL SAXPY( M, -CS*SN*AQOAP, $ A( 1, q ), 1, $ A( 1, p ), 1 ) IF( RSVEC ) THEN CALL SAXPY( MVL, T*APOAQ, $ V( 1, p ), 1, $ V( 1, q ), 1 ) CALL SAXPY( MVL, $ -CS*SN*AQOAP, $ V( 1, q ), 1, $ V( 1, p ), 1 ) END IF WORK( p ) = WORK( p ) / CS WORK( q ) = WORK( q )*CS ELSE IF( WORK( p ).GE.WORK( q ) ) $ THEN CALL SAXPY( M, -T*AQOAP, $ A( 1, q ), 1, $ A( 1, p ), 1 ) CALL SAXPY( M, CS*SN*APOAQ, $ A( 1, p ), 1, $ A( 1, q ), 1 ) WORK( p ) = WORK( p )*CS WORK( q ) = WORK( q ) / CS IF( RSVEC ) THEN CALL SAXPY( MVL, $ -T*AQOAP, $ V( 1, q ), 1, $ V( 1, p ), 1 ) CALL SAXPY( MVL, $ CS*SN*APOAQ, $ V( 1, p ), 1, $ V( 1, q ), 1 ) END IF ELSE CALL SAXPY( M, T*APOAQ, $ A( 1, p ), 1, $ A( 1, q ), 1 ) CALL SAXPY( M, $ -CS*SN*AQOAP, $ A( 1, q ), 1, $ A( 1, p ), 1 ) WORK( p ) = WORK( p ) / CS WORK( q ) = WORK( q )*CS IF( RSVEC ) THEN CALL SAXPY( MVL, $ T*APOAQ, V( 1, p ), $ 1, V( 1, q ), 1 ) CALL SAXPY( MVL, $ -CS*SN*AQOAP, $ V( 1, q ), 1, $ V( 1, p ), 1 ) END IF END IF END IF END IF END IF * ELSE IF( AAPP.GT.AAQQ ) THEN CALL SCOPY( M, A( 1, p ), 1, $ WORK( N+1 ), 1 ) CALL SLASCL( 'G', 0, 0, AAPP, ONE, $ M, 1, WORK( N+1 ), LDA, $ IERR ) CALL SLASCL( 'G', 0, 0, AAQQ, ONE, $ M, 1, A( 1, q ), LDA, $ IERR ) TEMP1 = -AAPQ*WORK( p ) / WORK( q ) CALL SAXPY( M, TEMP1, WORK( N+1 ), $ 1, A( 1, q ), 1 ) CALL SLASCL( 'G', 0, 0, ONE, AAQQ, $ M, 1, A( 1, q ), LDA, $ IERR ) SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, $ ONE-AAPQ*AAPQ ) ) MXSINJ = AMAX1( MXSINJ, SFMIN ) ELSE CALL SCOPY( M, A( 1, q ), 1, $ WORK( N+1 ), 1 ) CALL SLASCL( 'G', 0, 0, AAQQ, ONE, $ M, 1, WORK( N+1 ), LDA, $ IERR ) CALL SLASCL( 'G', 0, 0, AAPP, ONE, $ M, 1, A( 1, p ), LDA, $ IERR ) TEMP1 = -AAPQ*WORK( q ) / WORK( p ) CALL SAXPY( M, TEMP1, WORK( N+1 ), $ 1, A( 1, p ), 1 ) CALL SLASCL( 'G', 0, 0, ONE, AAPP, $ M, 1, A( 1, p ), LDA, $ IERR ) SVA( p ) = AAPP*SQRT( AMAX1( ZERO, $ ONE-AAPQ*AAPQ ) ) MXSINJ = AMAX1( MXSINJ, SFMIN ) END IF END IF * END IF ROTOK THEN ... ELSE * * In the case of cancellation in updating SVA(q) * .. recompute SVA(q) IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS ) $ THEN IF( ( AAQQ.LT.ROOTBIG ) .AND. $ ( AAQQ.GT.ROOTSFMIN ) ) THEN SVA( q ) = SNRM2( M, A( 1, q ), 1 )* $ WORK( q ) ELSE T = ZERO AAQQ = ONE CALL SLASSQ( M, A( 1, q ), 1, T, $ AAQQ ) SVA( q ) = T*SQRT( AAQQ )*WORK( q ) END IF END IF IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN IF( ( AAPP.LT.ROOTBIG ) .AND. $ ( AAPP.GT.ROOTSFMIN ) ) THEN AAPP = SNRM2( M, A( 1, p ), 1 )* $ WORK( p ) ELSE T = ZERO AAPP = ONE CALL SLASSQ( M, A( 1, p ), 1, T, $ AAPP ) AAPP = T*SQRT( AAPP )*WORK( p ) END IF SVA( p ) = AAPP END IF * end of OK rotation ELSE NOTROT = NOTROT + 1 *[RTD] SKIPPED = SKIPPED + 1 PSKIPPED = PSKIPPED + 1 IJBLSK = IJBLSK + 1 END IF ELSE NOTROT = NOTROT + 1 PSKIPPED = PSKIPPED + 1 IJBLSK = IJBLSK + 1 END IF * IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) ) $ THEN SVA( p ) = AAPP NOTROT = 0 GO TO 2011 END IF IF( ( i.LE.SWBAND ) .AND. $ ( PSKIPPED.GT.ROWSKIP ) ) THEN AAPP = -AAPP NOTROT = 0 GO TO 2203 END IF * 2200 CONTINUE * end of the q-loop 2203 CONTINUE * SVA( p ) = AAPP * ELSE * IF( AAPP.EQ.ZERO )NOTROT = NOTROT + $ MIN0( jgl+KBL-1, N ) - jgl + 1 IF( AAPP.LT.ZERO )NOTROT = 0 * END IF * 2100 CONTINUE * end of the p-loop 2010 CONTINUE * end of the jbc-loop 2011 CONTINUE *2011 bailed out of the jbc-loop DO 2012 p = igl, MIN0( igl+KBL-1, N ) SVA( p ) = ABS( SVA( p ) ) 2012 CONTINUE *** 2000 CONTINUE *2000 :: end of the ibr-loop * * .. update SVA(N) IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) ) $ THEN SVA( N ) = SNRM2( M, A( 1, N ), 1 )*WORK( N ) ELSE T = ZERO AAPP = ONE CALL SLASSQ( M, A( 1, N ), 1, T, AAPP ) SVA( N ) = T*SQRT( AAPP )*WORK( N ) END IF * * Additional steering devices * IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR. $ ( ISWROT.LE.N ) ) )SWBAND = i * IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.SQRT( FLOAT( N ) )* $ TOL ) .AND. ( FLOAT( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN GO TO 1994 END IF * IF( NOTROT.GE.EMPTSW )GO TO 1994 * 1993 CONTINUE * end i=1:NSWEEP loop * * #:( Reaching this point means that the procedure has not converged. INFO = NSWEEP - 1 GO TO 1995 * 1994 CONTINUE * #:) Reaching this point means numerical convergence after the i-th * sweep. * INFO = 0 * #:) INFO = 0 confirms successful iterations. 1995 CONTINUE * * Sort the singular values and find how many are above * the underflow threshold. * N2 = 0 N4 = 0 DO 5991 p = 1, N - 1 q = ISAMAX( N-p+1, SVA( p ), 1 ) + p - 1 IF( p.NE.q ) THEN TEMP1 = SVA( p ) SVA( p ) = SVA( q ) SVA( q ) = TEMP1 TEMP1 = WORK( p ) WORK( p ) = WORK( q ) WORK( q ) = TEMP1 CALL SSWAP( M, A( 1, p ), 1, A( 1, q ), 1 ) IF( RSVEC )CALL SSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 ) END IF IF( SVA( p ).NE.ZERO ) THEN N4 = N4 + 1 IF( SVA( p )*SKL.GT.SFMIN )N2 = N2 + 1 END IF 5991 CONTINUE IF( SVA( N ).NE.ZERO ) THEN N4 = N4 + 1 IF( SVA( N )*SKL.GT.SFMIN )N2 = N2 + 1 END IF * * Normalize the left singular vectors. * IF( LSVEC .OR. UCTOL ) THEN DO 1998 p = 1, N2 CALL SSCAL( M, WORK( p ) / SVA( p ), A( 1, p ), 1 ) 1998 CONTINUE END IF * * Scale the product of Jacobi rotations (assemble the fast rotations). * IF( RSVEC ) THEN IF( APPLV ) THEN DO 2398 p = 1, N CALL SSCAL( MVL, WORK( p ), V( 1, p ), 1 ) 2398 CONTINUE ELSE DO 2399 p = 1, N TEMP1 = ONE / SNRM2( MVL, V( 1, p ), 1 ) CALL SSCAL( MVL, TEMP1, V( 1, p ), 1 ) 2399 CONTINUE END IF END IF * * Undo scaling, if necessary (and possible). IF( ( ( SKL.GT.ONE ) .AND. ( SVA( 1 ).LT.( BIG / $ SKL ) ) ) .OR. ( ( SKL.LT.ONE ) .AND. ( SVA( N2 ).GT. $ ( SFMIN / SKL ) ) ) ) THEN DO 2400 p = 1, N SVA( p ) = SKL*SVA( p ) 2400 CONTINUE SKL = ONE END IF * WORK( 1 ) = SKL * The singular values of A are SKL*SVA(1:N). If SKL.NE.ONE * then some of the singular values may overflow or underflow and * the spectrum is given in this factored representation. * WORK( 2 ) = FLOAT( N4 ) * N4 is the number of computed nonzero singular values of A. * WORK( 3 ) = FLOAT( N2 ) * N2 is the number of singular values of A greater than SFMIN. * If N2 * that may carry some information. * WORK( 4 ) = FLOAT( i ) * i is the index of the last sweep before declaring convergence. * WORK( 5 ) = MXAAPQ * MXAAPQ is the largest absolute value of scaled pivots in the * last sweep * WORK( 6 ) = MXSINJ * MXSINJ is the largest absolute value of the sines of Jacobi angles * in the last sweep * RETURN * .. * .. END OF SGESVJ * .. END |