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SUBROUTINE DLASDQ( UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT, |
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$ U, LDU, C, LDC, WORK, INFO ) |
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* |
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* -- LAPACK auxiliary routine (version 3.2) -- |
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* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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* November 2006 |
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* |
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* .. Scalar Arguments .. |
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CHARACTER UPLO |
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INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU, SQRE |
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* .. |
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* .. Array Arguments .. |
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DOUBLE PRECISION C( LDC, * ), D( * ), E( * ), U( LDU, * ), |
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$ VT( LDVT, * ), WORK( * ) |
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* .. |
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* |
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* Purpose |
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* ======= |
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* |
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* DLASDQ computes the singular value decomposition (SVD) of a real |
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* (upper or lower) bidiagonal matrix with diagonal D and offdiagonal |
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* E, accumulating the transformations if desired. Letting B denote |
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* the input bidiagonal matrix, the algorithm computes orthogonal |
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* matrices Q and P such that B = Q * S * P' (P' denotes the transpose |
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* of P). The singular values S are overwritten on D. |
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* |
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* The input matrix U is changed to U * Q if desired. |
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* The input matrix VT is changed to P' * VT if desired. |
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* The input matrix C is changed to Q' * C if desired. |
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* |
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* See "Computing Small Singular Values of Bidiagonal Matrices With |
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* Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, |
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* LAPACK Working Note #3, for a detailed description of the algorithm. |
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* |
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* Arguments |
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* ========= |
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* |
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* UPLO (input) CHARACTER*1 |
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* On entry, UPLO specifies whether the input bidiagonal matrix |
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* is upper or lower bidiagonal, and wether it is square are |
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* not. |
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* UPLO = 'U' or 'u' B is upper bidiagonal. |
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* UPLO = 'L' or 'l' B is lower bidiagonal. |
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* |
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* SQRE (input) INTEGER |
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* = 0: then the input matrix is N-by-N. |
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* = 1: then the input matrix is N-by-(N+1) if UPLU = 'U' and |
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* (N+1)-by-N if UPLU = 'L'. |
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* |
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* The bidiagonal matrix has |
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* N = NL + NR + 1 rows and |
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* M = N + SQRE >= N columns. |
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* |
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* N (input) INTEGER |
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* On entry, N specifies the number of rows and columns |
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* in the matrix. N must be at least 0. |
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* |
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* NCVT (input) INTEGER |
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* On entry, NCVT specifies the number of columns of |
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* the matrix VT. NCVT must be at least 0. |
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* |
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* NRU (input) INTEGER |
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* On entry, NRU specifies the number of rows of |
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* the matrix U. NRU must be at least 0. |
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* |
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* NCC (input) INTEGER |
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* On entry, NCC specifies the number of columns of |
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* the matrix C. NCC must be at least 0. |
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* |
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* D (input/output) DOUBLE PRECISION array, dimension (N) |
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* On entry, D contains the diagonal entries of the |
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* bidiagonal matrix whose SVD is desired. On normal exit, |
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* D contains the singular values in ascending order. |
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* |
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* E (input/output) DOUBLE PRECISION array. |
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* dimension is (N-1) if SQRE = 0 and N if SQRE = 1. |
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* On entry, the entries of E contain the offdiagonal entries |
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* of the bidiagonal matrix whose SVD is desired. On normal |
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* exit, E will contain 0. If the algorithm does not converge, |
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* D and E will contain the diagonal and superdiagonal entries |
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* of a bidiagonal matrix orthogonally equivalent to the one |
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* given as input. |
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* |
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* VT (input/output) DOUBLE PRECISION array, dimension (LDVT, NCVT) |
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* On entry, contains a matrix which on exit has been |
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* premultiplied by P', dimension N-by-NCVT if SQRE = 0 |
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* and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0). |
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* |
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* LDVT (input) INTEGER |
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* On entry, LDVT specifies the leading dimension of VT as |
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* declared in the calling (sub) program. LDVT must be at |
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* least 1. If NCVT is nonzero LDVT must also be at least N. |
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* |
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* U (input/output) DOUBLE PRECISION array, dimension (LDU, N) |
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* On entry, contains a matrix which on exit has been |
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* postmultiplied by Q, dimension NRU-by-N if SQRE = 0 |
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* and NRU-by-(N+1) if SQRE = 1 (not referenced if NRU=0). |
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* |
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* LDU (input) INTEGER |
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* On entry, LDU specifies the leading dimension of U as |
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* declared in the calling (sub) program. LDU must be at |
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* least max( 1, NRU ) . |
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* |
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* C (input/output) DOUBLE PRECISION array, dimension (LDC, NCC) |
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* On entry, contains an N-by-NCC matrix which on exit |
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* has been premultiplied by Q' dimension N-by-NCC if SQRE = 0 |
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* and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0). |
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* |
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* LDC (input) INTEGER |
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* On entry, LDC specifies the leading dimension of C as |
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* declared in the calling (sub) program. LDC must be at |
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* least 1. If NCC is nonzero, LDC must also be at least N. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension (4*N) |
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* Workspace. Only referenced if one of NCVT, NRU, or NCC is |
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* nonzero, and if N is at least 2. |
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* |
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* INFO (output) INTEGER |
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* On exit, a value of 0 indicates a successful exit. |
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* If INFO < 0, argument number -INFO is illegal. |
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* If INFO > 0, the algorithm did not converge, and INFO |
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* specifies how many superdiagonals did not converge. |
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* |
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* Further Details |
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* =============== |
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* |
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* Based on contributions by |
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* Ming Gu and Huan Ren, Computer Science Division, University of |
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* California at Berkeley, USA |
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* |
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* ===================================================================== |
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* |
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* .. Parameters .. |
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DOUBLE PRECISION ZERO |
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PARAMETER ( ZERO = 0.0D+0 ) |
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* .. |
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* .. Local Scalars .. |
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LOGICAL ROTATE |
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INTEGER I, ISUB, IUPLO, J, NP1, SQRE1 |
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DOUBLE PRECISION CS, R, SMIN, SN |
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* .. |
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* .. External Subroutines .. |
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EXTERNAL DBDSQR, DLARTG, DLASR, DSWAP, XERBLA |
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* .. |
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* .. External Functions .. |
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LOGICAL LSAME |
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EXTERNAL LSAME |
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* .. |
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* .. Intrinsic Functions .. |
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INTRINSIC MAX |
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* .. |
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* .. Executable Statements .. |
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* |
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* Test the input parameters. |
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* |
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INFO = 0 |
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IUPLO = 0 |
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IF( LSAME( UPLO, 'U' ) ) |
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$ IUPLO = 1 |
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IF( LSAME( UPLO, 'L' ) ) |
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$ IUPLO = 2 |
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IF( IUPLO.EQ.0 ) THEN |
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INFO = -1 |
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ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN |
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INFO = -2 |
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ELSE IF( N.LT.0 ) THEN |
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INFO = -3 |
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ELSE IF( NCVT.LT.0 ) THEN |
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INFO = -4 |
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ELSE IF( NRU.LT.0 ) THEN |
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INFO = -5 |
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ELSE IF( NCC.LT.0 ) THEN |
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INFO = -6 |
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ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR. |
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$ ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN |
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INFO = -10 |
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ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN |
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INFO = -12 |
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ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR. |
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$ ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN |
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INFO = -14 |
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END IF |
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IF( INFO.NE.0 ) THEN |
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CALL XERBLA( 'DLASDQ', -INFO ) |
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RETURN |
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END IF |
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IF( N.EQ.0 ) |
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$ RETURN |
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* |
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* ROTATE is true if any singular vectors desired, false otherwise |
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* |
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ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 ) |
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NP1 = N + 1 |
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SQRE1 = SQRE |
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* |
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* If matrix non-square upper bidiagonal, rotate to be lower |
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* bidiagonal. The rotations are on the right. |
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* |
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IF( ( IUPLO.EQ.1 ) .AND. ( SQRE1.EQ.1 ) ) THEN |
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DO 10 I = 1, N - 1 |
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CALL DLARTG( D( I ), E( I ), CS, SN, R ) |
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D( I ) = R |
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E( I ) = SN*D( I+1 ) |
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D( I+1 ) = CS*D( I+1 ) |
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IF( ROTATE ) THEN |
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WORK( I ) = CS |
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WORK( N+I ) = SN |
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END IF |
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10 CONTINUE |
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CALL DLARTG( D( N ), E( N ), CS, SN, R ) |
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D( N ) = R |
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E( N ) = ZERO |
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IF( ROTATE ) THEN |
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WORK( N ) = CS |
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WORK( N+N ) = SN |
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END IF |
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IUPLO = 2 |
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SQRE1 = 0 |
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* |
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* Update singular vectors if desired. |
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* |
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IF( NCVT.GT.0 ) |
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$ CALL DLASR( 'L', 'V', 'F', NP1, NCVT, WORK( 1 ), |
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$ WORK( NP1 ), VT, LDVT ) |
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END IF |
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* |
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* If matrix lower bidiagonal, rotate to be upper bidiagonal |
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* by applying Givens rotations on the left. |
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* |
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IF( IUPLO.EQ.2 ) THEN |
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DO 20 I = 1, N - 1 |
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CALL DLARTG( D( I ), E( I ), CS, SN, R ) |
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D( I ) = R |
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E( I ) = SN*D( I+1 ) |
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D( I+1 ) = CS*D( I+1 ) |
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IF( ROTATE ) THEN |
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WORK( I ) = CS |
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WORK( N+I ) = SN |
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END IF |
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20 CONTINUE |
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* |
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* If matrix (N+1)-by-N lower bidiagonal, one additional |
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* rotation is needed. |
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* |
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IF( SQRE1.EQ.1 ) THEN |
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CALL DLARTG( D( N ), E( N ), CS, SN, R ) |
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D( N ) = R |
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IF( ROTATE ) THEN |
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WORK( N ) = CS |
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WORK( N+N ) = SN |
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END IF |
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END IF |
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* |
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* Update singular vectors if desired. |
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* |
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IF( NRU.GT.0 ) THEN |
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IF( SQRE1.EQ.0 ) THEN |
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CALL DLASR( 'R', 'V', 'F', NRU, N, WORK( 1 ), |
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$ WORK( NP1 ), U, LDU ) |
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ELSE |
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CALL DLASR( 'R', 'V', 'F', NRU, NP1, WORK( 1 ), |
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$ WORK( NP1 ), U, LDU ) |
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END IF |
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END IF |
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IF( NCC.GT.0 ) THEN |
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IF( SQRE1.EQ.0 ) THEN |
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CALL DLASR( 'L', 'V', 'F', N, NCC, WORK( 1 ), |
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$ WORK( NP1 ), C, LDC ) |
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ELSE |
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CALL DLASR( 'L', 'V', 'F', NP1, NCC, WORK( 1 ), |
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$ WORK( NP1 ), C, LDC ) |
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END IF |
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END IF |
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END IF |
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* |
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* Call DBDSQR to compute the SVD of the reduced real |
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* N-by-N upper bidiagonal matrix. |
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* |
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CALL DBDSQR( 'U', N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, |
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$ LDC, WORK, INFO ) |
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* |
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* Sort the singular values into ascending order (insertion sort on |
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* singular values, but only one transposition per singular vector) |
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* |
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DO 40 I = 1, N |
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* |
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* Scan for smallest D(I). |
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* |
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ISUB = I |
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SMIN = D( I ) |
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DO 30 J = I + 1, N |
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IF( D( J ).LT.SMIN ) THEN |
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ISUB = J |
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SMIN = D( J ) |
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END IF |
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30 CONTINUE |
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IF( ISUB.NE.I ) THEN |
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* |
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* Swap singular values and vectors. |
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* |
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D( ISUB ) = D( I ) |
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D( I ) = SMIN |
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IF( NCVT.GT.0 ) |
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$ CALL DSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( I, 1 ), LDVT ) |
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IF( NRU.GT.0 ) |
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$ CALL DSWAP( NRU, U( 1, ISUB ), 1, U( 1, I ), 1 ) |
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IF( NCC.GT.0 ) |
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$ CALL DSWAP( NCC, C( ISUB, 1 ), LDC, C( I, 1 ), LDC ) |
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END IF |
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40 CONTINUE |
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* |
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RETURN |
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* |
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* End of DLASDQ |
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* |
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END |