SUBROUTINE SLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, $ LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, $ IDXC, IDXQ, COLTYP, INFO ) * * -- LAPACK auxiliary routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE REAL ALPHA, BETA * .. * .. Array Arguments .. INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ), $ IDXQ( * ) REAL D( * ), DSIGMA( * ), U( LDU, * ), $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), $ Z( * ) * .. * * Purpose * ======= * * SLASD2 merges the two sets of singular values together into a single * sorted set. Then it tries to deflate the size of the problem. * There are two ways in which deflation can occur: when two or more * singular values are close together or if there is a tiny entry in the * Z vector. For each such occurrence the order of the related secular * equation problem is reduced by one. * * SLASD2 is called from SLASD1. * * Arguments * ========= * * NL (input) INTEGER * The row dimension of the upper block. NL >= 1. * * NR (input) INTEGER * The row dimension of the lower block. NR >= 1. * * SQRE (input) INTEGER * = 0: the lower block is an NR-by-NR square matrix. * = 1: the lower block is an NR-by-(NR+1) rectangular matrix. * * The bidiagonal matrix has N = NL + NR + 1 rows and * M = N + SQRE >= N columns. * * K (output) INTEGER * Contains the dimension of the non-deflated matrix, * This is the order of the related secular equation. 1 <= K <=N. * * D (input/output) REAL array, dimension (N) * On entry D contains the singular values of the two submatrices * to be combined. On exit D contains the trailing (N-K) updated * singular values (those which were deflated) sorted into * increasing order. * * Z (output) REAL array, dimension (N) * On exit Z contains the updating row vector in the secular * equation. * * ALPHA (input) REAL * Contains the diagonal element associated with the added row. * * BETA (input) REAL * Contains the off-diagonal element associated with the added * row. * * U (input/output) REAL array, dimension (LDU,N) * On entry U contains the left singular vectors of two * submatrices in the two square blocks with corners at (1,1), * (NL, NL), and (NL+2, NL+2), (N,N). * On exit U contains the trailing (N-K) updated left singular * vectors (those which were deflated) in its last N-K columns. * * LDU (input) INTEGER * The leading dimension of the array U. LDU >= N. * * VT (input/output) REAL array, dimension (LDVT,M) * On entry VT' contains the right singular vectors of two * submatrices in the two square blocks with corners at (1,1), * (NL+1, NL+1), and (NL+2, NL+2), (M,M). * On exit VT' contains the trailing (N-K) updated right singular * vectors (those which were deflated) in its last N-K columns. * In case SQRE =1, the last row of VT spans the right null * space. * * LDVT (input) INTEGER * The leading dimension of the array VT. LDVT >= M. * * DSIGMA (output) REAL array, dimension (N) * Contains a copy of the diagonal elements (K-1 singular values * and one zero) in the secular equation. * * U2 (output) REAL array, dimension (LDU2,N) * Contains a copy of the first K-1 left singular vectors which * will be used by SLASD3 in a matrix multiply (SGEMM) to solve * for the new left singular vectors. U2 is arranged into four * blocks. The first block contains a column with 1 at NL+1 and * zero everywhere else; the second block contains non-zero * entries only at and above NL; the third contains non-zero * entries only below NL+1; and the fourth is dense. * * LDU2 (input) INTEGER * The leading dimension of the array U2. LDU2 >= N. * * VT2 (output) REAL array, dimension (LDVT2,N) * VT2' contains a copy of the first K right singular vectors * which will be used by SLASD3 in a matrix multiply (SGEMM) to * solve for the new right singular vectors. VT2 is arranged into * three blocks. The first block contains a row that corresponds * to the special 0 diagonal element in SIGMA; the second block * contains non-zeros only at and before NL +1; the third block * contains non-zeros only at and after NL +2. * * LDVT2 (input) INTEGER * The leading dimension of the array VT2. LDVT2 >= M. * * IDXP (workspace) INTEGER array, dimension (N) * This will contain the permutation used to place deflated * values of D at the end of the array. On output IDXP(2:K) * points to the nondeflated D-values and IDXP(K+1:N) * points to the deflated singular values. * * IDX (workspace) INTEGER array, dimension (N) * This will contain the permutation used to sort the contents of * D into ascending order. * * IDXC (output) INTEGER array, dimension (N) * This will contain the permutation used to arrange the columns * of the deflated U matrix into three groups: the first group * contains non-zero entries only at and above NL, the second * contains non-zero entries only below NL+2, and the third is * dense. * * IDXQ (input/output) INTEGER array, dimension (N) * This contains the permutation which separately sorts the two * sub-problems in D into ascending order. Note that entries in * the first hlaf of this permutation must first be moved one * position backward; and entries in the second half * must first have NL+1 added to their values. * * COLTYP (workspace/output) INTEGER array, dimension (N) * As workspace, this will contain a label which will indicate * which of the following types a column in the U2 matrix or a * row in the VT2 matrix is: * 1 : non-zero in the upper half only * 2 : non-zero in the lower half only * 3 : dense * 4 : deflated * * On exit, it is an array of dimension 4, with COLTYP(I) being * the dimension of the I-th type columns. * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * * Further Details * =============== * * Based on contributions by * Ming Gu and Huan Ren, Computer Science Division, University of * California at Berkeley, USA * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO, EIGHT PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0, $ EIGHT = 8.0E+0 ) * .. * .. Local Arrays .. INTEGER CTOT( 4 ), PSM( 4 ) * .. * .. Local Scalars .. INTEGER CT, I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M, $ N, NLP1, NLP2 REAL C, EPS, HLFTOL, S, TAU, TOL, Z1 * .. * .. External Functions .. REAL SLAMCH, SLAPY2 EXTERNAL SLAMCH, SLAPY2 * .. * .. External Subroutines .. EXTERNAL SCOPY, SLACPY, SLAMRG, SLASET, SROT, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( NL.LT.1 ) THEN INFO = -1 ELSE IF( NR.LT.1 ) THEN INFO = -2 ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN INFO = -3 END IF * N = NL + NR + 1 M = N + SQRE * IF( LDU.LT.N ) THEN INFO = -10 ELSE IF( LDVT.LT.M ) THEN INFO = -12 ELSE IF( LDU2.LT.N ) THEN INFO = -15 ELSE IF( LDVT2.LT.M ) THEN INFO = -17 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLASD2', -INFO ) RETURN END IF * NLP1 = NL + 1 NLP2 = NL + 2 * * Generate the first part of the vector Z; and move the singular * values in the first part of D one position backward. * Z1 = ALPHA*VT( NLP1, NLP1 ) Z( 1 ) = Z1 DO 10 I = NL, 1, -1 Z( I+1 ) = ALPHA*VT( I, NLP1 ) D( I+1 ) = D( I ) IDXQ( I+1 ) = IDXQ( I ) + 1 10 CONTINUE * * Generate the second part of the vector Z. * DO 20 I = NLP2, M Z( I ) = BETA*VT( I, NLP2 ) 20 CONTINUE * * Initialize some reference arrays. * DO 30 I = 2, NLP1 COLTYP( I ) = 1 30 CONTINUE DO 40 I = NLP2, N COLTYP( I ) = 2 40 CONTINUE * * Sort the singular values into increasing order * DO 50 I = NLP2, N IDXQ( I ) = IDXQ( I ) + NLP1 50 CONTINUE * * DSIGMA, IDXC, IDXC, and the first column of U2 * are used as storage space. * DO 60 I = 2, N DSIGMA( I ) = D( IDXQ( I ) ) U2( I, 1 ) = Z( IDXQ( I ) ) IDXC( I ) = COLTYP( IDXQ( I ) ) 60 CONTINUE * CALL SLAMRG( NL, NR, DSIGMA( 2 ), 1, 1, IDX( 2 ) ) * DO 70 I = 2, N IDXI = 1 + IDX( I ) D( I ) = DSIGMA( IDXI ) Z( I ) = U2( IDXI, 1 ) COLTYP( I ) = IDXC( IDXI ) 70 CONTINUE * * Calculate the allowable deflation tolerance * EPS = SLAMCH( 'Epsilon' ) TOL = MAX( ABS( ALPHA ), ABS( BETA ) ) TOL = EIGHT*EPS*MAX( ABS( D( N ) ), TOL ) * * There are 2 kinds of deflation -- first a value in the z-vector * is small, second two (or more) singular values are very close * together (their difference is small). * * If the value in the z-vector is small, we simply permute the * array so that the corresponding singular value is moved to the * end. * * If two values in the D-vector are close, we perform a two-sided * rotation designed to make one of the corresponding z-vector * entries zero, and then permute the array so that the deflated * singular value is moved to the end. * * If there are multiple singular values then the problem deflates. * Here the number of equal singular values are found. As each equal * singular value is found, an elementary reflector is computed to * rotate the corresponding singular subspace so that the * corresponding components of Z are zero in this new basis. * K = 1 K2 = N + 1 DO 80 J = 2, N IF( ABS( Z( J ) ).LE.TOL ) THEN * * Deflate due to small z component. * K2 = K2 - 1 IDXP( K2 ) = J COLTYP( J ) = 4 IF( J.EQ.N ) $ GO TO 120 ELSE JPREV = J GO TO 90 END IF 80 CONTINUE 90 CONTINUE J = JPREV 100 CONTINUE J = J + 1 IF( J.GT.N ) $ GO TO 110 IF( ABS( Z( J ) ).LE.TOL ) THEN * * Deflate due to small z component. * K2 = K2 - 1 IDXP( K2 ) = J COLTYP( J ) = 4 ELSE * * Check if singular values are close enough to allow deflation. * IF( ABS( D( J )-D( JPREV ) ).LE.TOL ) THEN * * Deflation is possible. * S = Z( JPREV ) C = Z( J ) * * Find sqrt(a**2+b**2) without overflow or * destructive underflow. * TAU = SLAPY2( C, S ) C = C / TAU S = -S / TAU Z( J ) = TAU Z( JPREV ) = ZERO * * Apply back the Givens rotation to the left and right * singular vector matrices. * IDXJP = IDXQ( IDX( JPREV )+1 ) IDXJ = IDXQ( IDX( J )+1 ) IF( IDXJP.LE.NLP1 ) THEN IDXJP = IDXJP - 1 END IF IF( IDXJ.LE.NLP1 ) THEN IDXJ = IDXJ - 1 END IF CALL SROT( N, U( 1, IDXJP ), 1, U( 1, IDXJ ), 1, C, S ) CALL SROT( M, VT( IDXJP, 1 ), LDVT, VT( IDXJ, 1 ), LDVT, C, $ S ) IF( COLTYP( J ).NE.COLTYP( JPREV ) ) THEN COLTYP( J ) = 3 END IF COLTYP( JPREV ) = 4 K2 = K2 - 1 IDXP( K2 ) = JPREV JPREV = J ELSE K = K + 1 U2( K, 1 ) = Z( JPREV ) DSIGMA( K ) = D( JPREV ) IDXP( K ) = JPREV JPREV = J END IF END IF GO TO 100 110 CONTINUE * * Record the last singular value. * K = K + 1 U2( K, 1 ) = Z( JPREV ) DSIGMA( K ) = D( JPREV ) IDXP( K ) = JPREV * 120 CONTINUE * * Count up the total number of the various types of columns, then * form a permutation which positions the four column types into * four groups of uniform structure (although one or more of these * groups may be empty). * DO 130 J = 1, 4 CTOT( J ) = 0 130 CONTINUE DO 140 J = 2, N CT = COLTYP( J ) CTOT( CT ) = CTOT( CT ) + 1 140 CONTINUE * * PSM(*) = Position in SubMatrix (of types 1 through 4) * PSM( 1 ) = 2 PSM( 2 ) = 2 + CTOT( 1 ) PSM( 3 ) = PSM( 2 ) + CTOT( 2 ) PSM( 4 ) = PSM( 3 ) + CTOT( 3 ) * * Fill out the IDXC array so that the permutation which it induces * will place all type-1 columns first, all type-2 columns next, * then all type-3's, and finally all type-4's, starting from the * second column. This applies similarly to the rows of VT. * DO 150 J = 2, N JP = IDXP( J ) CT = COLTYP( JP ) IDXC( PSM( CT ) ) = J PSM( CT ) = PSM( CT ) + 1 150 CONTINUE * * Sort the singular values and corresponding singular vectors into * DSIGMA, U2, and VT2 respectively. The singular values/vectors * which were not deflated go into the first K slots of DSIGMA, U2, * and VT2 respectively, while those which were deflated go into the * last N - K slots, except that the first column/row will be treated * separately. * DO 160 J = 2, N JP = IDXP( J ) DSIGMA( J ) = D( JP ) IDXJ = IDXQ( IDX( IDXP( IDXC( J ) ) )+1 ) IF( IDXJ.LE.NLP1 ) THEN IDXJ = IDXJ - 1 END IF CALL SCOPY( N, U( 1, IDXJ ), 1, U2( 1, J ), 1 ) CALL SCOPY( M, VT( IDXJ, 1 ), LDVT, VT2( J, 1 ), LDVT2 ) 160 CONTINUE * * Determine DSIGMA(1), DSIGMA(2) and Z(1) * DSIGMA( 1 ) = ZERO HLFTOL = TOL / TWO IF( ABS( DSIGMA( 2 ) ).LE.HLFTOL ) $ DSIGMA( 2 ) = HLFTOL IF( M.GT.N ) THEN Z( 1 ) = SLAPY2( Z1, Z( M ) ) IF( Z( 1 ).LE.TOL ) THEN C = ONE S = ZERO Z( 1 ) = TOL ELSE C = Z1 / Z( 1 ) S = Z( M ) / Z( 1 ) END IF ELSE IF( ABS( Z1 ).LE.TOL ) THEN Z( 1 ) = TOL ELSE Z( 1 ) = Z1 END IF END IF * * Move the rest of the updating row to Z. * CALL SCOPY( K-1, U2( 2, 1 ), 1, Z( 2 ), 1 ) * * Determine the first column of U2, the first row of VT2 and the * last row of VT. * CALL SLASET( 'A', N, 1, ZERO, ZERO, U2, LDU2 ) U2( NLP1, 1 ) = ONE IF( M.GT.N ) THEN DO 170 I = 1, NLP1 VT( M, I ) = -S*VT( NLP1, I ) VT2( 1, I ) = C*VT( NLP1, I ) 170 CONTINUE DO 180 I = NLP2, M VT2( 1, I ) = S*VT( M, I ) VT( M, I ) = C*VT( M, I ) 180 CONTINUE ELSE CALL SCOPY( M, VT( NLP1, 1 ), LDVT, VT2( 1, 1 ), LDVT2 ) END IF IF( M.GT.N ) THEN CALL SCOPY( M, VT( M, 1 ), LDVT, VT2( M, 1 ), LDVT2 ) END IF * * The deflated singular values and their corresponding vectors go * into the back of D, U, and V respectively. * IF( N.GT.K ) THEN CALL SCOPY( N-K, DSIGMA( K+1 ), 1, D( K+1 ), 1 ) CALL SLACPY( 'A', N, N-K, U2( 1, K+1 ), LDU2, U( 1, K+1 ), $ LDU ) CALL SLACPY( 'A', N-K, M, VT2( K+1, 1 ), LDVT2, VT( K+1, 1 ), $ LDVT ) END IF * * Copy CTOT into COLTYP for referencing in SLASD3. * DO 190 J = 1, 4 COLTYP( J ) = CTOT( J ) 190 CONTINUE * RETURN * * End of SLASD2 * END