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SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, |
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$ LDU, C, LDC, WORK, INFO ) |
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* |
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* -- LAPACK 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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* January 2007 |
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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 |
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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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* DBDSQR computes the singular values and, optionally, the right and/or |
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* left singular vectors from the singular value decomposition (SVD) of |
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* a real N-by-N (upper or lower) bidiagonal matrix B using the implicit |
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* zero-shift QR algorithm. The SVD of B has the form |
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* |
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* B = Q * S * P**T |
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* |
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* where S is the diagonal matrix of singular values, Q is an orthogonal |
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* matrix of left singular vectors, and P is an orthogonal matrix of |
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* right singular vectors. If left singular vectors are requested, this |
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* subroutine actually returns U*Q instead of Q, and, if right singular |
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* vectors are requested, this subroutine returns P**T*VT instead of |
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* P**T, for given real input matrices U and VT. When U and VT are the |
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* orthogonal matrices that reduce a general matrix A to bidiagonal |
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* form: A = U*B*VT, as computed by DGEBRD, then |
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* |
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* A = (U*Q) * S * (P**T*VT) |
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* |
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* is the SVD of A. Optionally, the subroutine may also compute Q**T*C |
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* for a given real input matrix C. |
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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 (or SIAM J. Sci. Statist. Comput. vol. 11, |
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* no. 5, pp. 873-912, Sept 1990) and |
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* "Accurate singular values and differential qd algorithms," by |
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* B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics |
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* Department, University of California at Berkeley, July 1992 |
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* 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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* = 'U': B is upper bidiagonal; |
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* = 'L': B is lower bidiagonal. |
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* |
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* N (input) INTEGER |
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* The order of the matrix B. N >= 0. |
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* |
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* NCVT (input) INTEGER |
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* The number of columns of the matrix VT. NCVT >= 0. |
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* |
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* NRU (input) INTEGER |
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* The number of rows of the matrix U. NRU >= 0. |
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* |
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* NCC (input) INTEGER |
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* The number of columns of the matrix C. NCC >= 0. |
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* |
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* D (input/output) DOUBLE PRECISION array, dimension (N) |
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* On entry, the n diagonal elements of the bidiagonal matrix B. |
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* On exit, if INFO=0, the singular values of B in decreasing |
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* order. |
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* |
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* E (input/output) DOUBLE PRECISION array, dimension (N-1) |
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* On entry, the N-1 offdiagonal elements of the bidiagonal |
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* matrix B. |
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* On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E |
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* will contain the diagonal and superdiagonal elements of a |
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* bidiagonal matrix orthogonally equivalent to the one given |
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* 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, an N-by-NCVT matrix VT. |
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* On exit, VT is overwritten by P**T * VT. |
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* Not referenced if NCVT = 0. |
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* |
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* LDVT (input) INTEGER |
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* The leading dimension of the array VT. |
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* LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. |
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* |
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* U (input/output) DOUBLE PRECISION array, dimension (LDU, N) |
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* On entry, an NRU-by-N matrix U. |
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* On exit, U is overwritten by U * Q. |
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* Not referenced if NRU = 0. |
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* |
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* LDU (input) INTEGER |
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* The leading dimension of the array U. LDU >= 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, an N-by-NCC matrix C. |
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* On exit, C is overwritten by Q**T * C. |
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* Not referenced if NCC = 0. |
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* |
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* LDC (input) INTEGER |
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* The leading dimension of the array C. |
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* LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension (4*N) |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: If INFO = -i, the i-th argument had an illegal value |
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* > 0: |
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* if NCVT = NRU = NCC = 0, |
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* = 1, a split was marked by a positive value in E |
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* = 2, current block of Z not diagonalized after 30*N |
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* iterations (in inner while loop) |
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* = 3, termination criterion of outer while loop not met |
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* (program created more than N unreduced blocks) |
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* else NCVT = NRU = NCC = 0, |
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* the algorithm did not converge; D and E contain the |
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* elements of a bidiagonal matrix which is orthogonally |
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* similar to the input matrix B; if INFO = i, i |
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* elements of E have not converged to zero. |
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* |
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* Internal Parameters |
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* =================== |
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* |
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* TOLMUL DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8))) |
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* TOLMUL controls the convergence criterion of the QR loop. |
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* If it is positive, TOLMUL*EPS is the desired relative |
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* precision in the computed singular values. |
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* If it is negative, abs(TOLMUL*EPS*sigma_max) is the |
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* desired absolute accuracy in the computed singular |
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* values (corresponds to relative accuracy |
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* abs(TOLMUL*EPS) in the largest singular value. |
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* abs(TOLMUL) should be between 1 and 1/EPS, and preferably |
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* between 10 (for fast convergence) and .1/EPS |
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* (for there to be some accuracy in the results). |
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* Default is to lose at either one eighth or 2 of the |
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* available decimal digits in each computed singular value |
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* (whichever is smaller). |
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* |
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* MAXITR INTEGER, default = 6 |
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* MAXITR controls the maximum number of passes of the |
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* algorithm through its inner loop. The algorithms stops |
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* (and so fails to converge) if the number of passes |
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* through the inner loop exceeds MAXITR*N**2. |
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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 = 0.0D0 ) |
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DOUBLE PRECISION ONE |
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PARAMETER ( ONE = 1.0D0 ) |
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DOUBLE PRECISION NEGONE |
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PARAMETER ( NEGONE = -1.0D0 ) |
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DOUBLE PRECISION HNDRTH |
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PARAMETER ( HNDRTH = 0.01D0 ) |
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DOUBLE PRECISION TEN |
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PARAMETER ( TEN = 10.0D0 ) |
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DOUBLE PRECISION HNDRD |
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PARAMETER ( HNDRD = 100.0D0 ) |
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DOUBLE PRECISION MEIGTH |
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PARAMETER ( MEIGTH = -0.125D0 ) |
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INTEGER MAXITR |
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PARAMETER ( MAXITR = 6 ) |
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* .. |
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* .. Local Scalars .. |
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LOGICAL LOWER, ROTATE |
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INTEGER I, IDIR, ISUB, ITER, J, LL, LLL, M, MAXIT, NM1, |
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$ NM12, NM13, OLDLL, OLDM |
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DOUBLE PRECISION ABSE, ABSS, COSL, COSR, CS, EPS, F, G, H, MU, |
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$ OLDCS, OLDSN, R, SHIFT, SIGMN, SIGMX, SINL, |
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$ SINR, SLL, SMAX, SMIN, SMINL, SMINOA, |
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$ SN, THRESH, TOL, TOLMUL, UNFL |
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* .. |
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* .. External Functions .. |
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LOGICAL LSAME |
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DOUBLE PRECISION DLAMCH |
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EXTERNAL LSAME, DLAMCH |
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* .. |
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* .. External Subroutines .. |
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EXTERNAL DLARTG, DLAS2, DLASQ1, DLASR, DLASV2, DROT, |
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$ DSCAL, DSWAP, XERBLA |
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* .. |
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* .. Intrinsic Functions .. |
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INTRINSIC ABS, DBLE, MAX, MIN, SIGN, SQRT |
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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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LOWER = LSAME( UPLO, 'L' ) |
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IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LOWER ) THEN |
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INFO = -1 |
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ELSE IF( N.LT.0 ) THEN |
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INFO = -2 |
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ELSE IF( NCVT.LT.0 ) THEN |
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INFO = -3 |
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ELSE IF( NRU.LT.0 ) THEN |
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INFO = -4 |
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ELSE IF( NCC.LT.0 ) THEN |
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INFO = -5 |
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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 = -9 |
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ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN |
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INFO = -11 |
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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 = -13 |
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END IF |
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IF( INFO.NE.0 ) THEN |
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CALL XERBLA( 'DBDSQR', -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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IF( N.EQ.1 ) |
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$ GO TO 160 |
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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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* |
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* If no singular vectors desired, use qd algorithm |
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* |
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IF( .NOT.ROTATE ) THEN |
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CALL DLASQ1( N, D, E, WORK, INFO ) |
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RETURN |
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END IF |
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* |
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NM1 = N - 1 |
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NM12 = NM1 + NM1 |
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NM13 = NM12 + NM1 |
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IDIR = 0 |
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* |
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* Get machine constants |
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* |
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EPS = DLAMCH( 'Epsilon' ) |
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UNFL = DLAMCH( 'Safe minimum' ) |
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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( LOWER ) 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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WORK( I ) = CS |
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WORK( NM1+I ) = SN |
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10 CONTINUE |
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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 ) |
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$ CALL DLASR( 'R', 'V', 'F', NRU, N, WORK( 1 ), WORK( N ), U, |
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$ LDU ) |
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IF( NCC.GT.0 ) |
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$ CALL DLASR( 'L', 'V', 'F', N, NCC, WORK( 1 ), WORK( N ), C, |
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$ LDC ) |
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END IF |
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* |
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* Compute singular values to relative accuracy TOL |
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* (By setting TOL to be negative, algorithm will compute |
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* singular values to absolute accuracy ABS(TOL)*norm(input matrix)) |
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* |
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TOLMUL = MAX( TEN, MIN( HNDRD, EPS**MEIGTH ) ) |
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TOL = TOLMUL*EPS |
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* |
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* Compute approximate maximum, minimum singular values |
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* |
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SMAX = ZERO |
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DO 20 I = 1, N |
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SMAX = MAX( SMAX, ABS( D( I ) ) ) |
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20 CONTINUE |
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DO 30 I = 1, N - 1 |
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SMAX = MAX( SMAX, ABS( E( I ) ) ) |
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30 CONTINUE |
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SMINL = ZERO |
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IF( TOL.GE.ZERO ) THEN |
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* |
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* Relative accuracy desired |
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* |
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SMINOA = ABS( D( 1 ) ) |
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IF( SMINOA.EQ.ZERO ) |
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$ GO TO 50 |
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MU = SMINOA |
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DO 40 I = 2, N |
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MU = ABS( D( I ) )*( MU / ( MU+ABS( E( I-1 ) ) ) ) |
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SMINOA = MIN( SMINOA, MU ) |
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IF( SMINOA.EQ.ZERO ) |
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$ GO TO 50 |
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40 CONTINUE |
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50 CONTINUE |
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SMINOA = SMINOA / SQRT( DBLE( N ) ) |
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THRESH = MAX( TOL*SMINOA, MAXITR*N*N*UNFL ) |
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ELSE |
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* |
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* Absolute accuracy desired |
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* |
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THRESH = MAX( ABS( TOL )*SMAX, MAXITR*N*N*UNFL ) |
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END IF |
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* |
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* Prepare for main iteration loop for the singular values |
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* (MAXIT is the maximum number of passes through the inner |
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* loop permitted before nonconvergence signalled.) |
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* |
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MAXIT = MAXITR*N*N |
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ITER = 0 |
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OLDLL = -1 |
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OLDM = -1 |
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* |
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* M points to last element of unconverged part of matrix |
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* |
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M = N |
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* |
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* Begin main iteration loop |
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* |
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60 CONTINUE |
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* |
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* Check for convergence or exceeding iteration count |
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* |
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IF( M.LE.1 ) |
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$ GO TO 160 |
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IF( ITER.GT.MAXIT ) |
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$ GO TO 200 |
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* |
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* Find diagonal block of matrix to work on |
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* |
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IF( TOL.LT.ZERO .AND. ABS( D( M ) ).LE.THRESH ) |
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$ D( M ) = ZERO |
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SMAX = ABS( D( M ) ) |
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SMIN = SMAX |
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DO 70 LLL = 1, M - 1 |
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LL = M - LLL |
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ABSS = ABS( D( LL ) ) |
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ABSE = ABS( E( LL ) ) |
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IF( TOL.LT.ZERO .AND. ABSS.LE.THRESH ) |
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$ D( LL ) = ZERO |
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IF( ABSE.LE.THRESH ) |
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$ GO TO 80 |
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SMIN = MIN( SMIN, ABSS ) |
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SMAX = MAX( SMAX, ABSS, ABSE ) |
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70 CONTINUE |
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LL = 0 |
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GO TO 90 |
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80 CONTINUE |
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E( LL ) = ZERO |
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* |
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* Matrix splits since E(LL) = 0 |
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* |
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IF( LL.EQ.M-1 ) THEN |
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* |
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* Convergence of bottom singular value, return to top of loop |
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* |
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M = M - 1 |
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GO TO 60 |
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END IF |
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90 CONTINUE |
| 367 |
LL = LL + 1 |
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* |
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* E(LL) through E(M-1) are nonzero, E(LL-1) is zero |
| 370 |
* |
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IF( LL.EQ.M-1 ) THEN |
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* |
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* 2 by 2 block, handle separately |
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* |
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CALL DLASV2( D( M-1 ), E( M-1 ), D( M ), SIGMN, SIGMX, SINR, |
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$ COSR, SINL, COSL ) |
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D( M-1 ) = SIGMX |
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E( M-1 ) = ZERO |
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D( M ) = SIGMN |
| 380 |
* |
| 381 |
* Compute singular vectors, if desired |
| 382 |
* |
| 383 |
IF( NCVT.GT.0 ) |
| 384 |
$ CALL DROT( NCVT, VT( M-1, 1 ), LDVT, VT( M, 1 ), LDVT, COSR, |
| 385 |
$ SINR ) |
| 386 |
IF( NRU.GT.0 ) |
| 387 |
$ CALL DROT( NRU, U( 1, M-1 ), 1, U( 1, M ), 1, COSL, SINL ) |
| 388 |
IF( NCC.GT.0 ) |
| 389 |
$ CALL DROT( NCC, C( M-1, 1 ), LDC, C( M, 1 ), LDC, COSL, |
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$ SINL ) |
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M = M - 2 |
| 392 |
GO TO 60 |
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END IF |
| 394 |
* |
| 395 |
* If working on new submatrix, choose shift direction |
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* (from larger end diagonal element towards smaller) |
| 397 |
* |
| 398 |
IF( LL.GT.OLDM .OR. M.LT.OLDLL ) THEN |
| 399 |
IF( ABS( D( LL ) ).GE.ABS( D( M ) ) ) THEN |
| 400 |
* |
| 401 |
* Chase bulge from top (big end) to bottom (small end) |
| 402 |
* |
| 403 |
IDIR = 1 |
| 404 |
ELSE |
| 405 |
* |
| 406 |
* Chase bulge from bottom (big end) to top (small end) |
| 407 |
* |
| 408 |
IDIR = 2 |
| 409 |
END IF |
| 410 |
END IF |
| 411 |
* |
| 412 |
* Apply convergence tests |
| 413 |
* |
| 414 |
IF( IDIR.EQ.1 ) THEN |
| 415 |
* |
| 416 |
* Run convergence test in forward direction |
| 417 |
* First apply standard test to bottom of matrix |
| 418 |
* |
| 419 |
IF( ABS( E( M-1 ) ).LE.ABS( TOL )*ABS( D( M ) ) .OR. |
| 420 |
$ ( TOL.LT.ZERO .AND. ABS( E( M-1 ) ).LE.THRESH ) ) THEN |
| 421 |
E( M-1 ) = ZERO |
| 422 |
GO TO 60 |
| 423 |
END IF |
| 424 |
* |
| 425 |
IF( TOL.GE.ZERO ) THEN |
| 426 |
* |
| 427 |
* If relative accuracy desired, |
| 428 |
* apply convergence criterion forward |
| 429 |
* |
| 430 |
MU = ABS( D( LL ) ) |
| 431 |
SMINL = MU |
| 432 |
DO 100 LLL = LL, M - 1 |
| 433 |
IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN |
| 434 |
E( LLL ) = ZERO |
| 435 |
GO TO 60 |
| 436 |
END IF |
| 437 |
MU = ABS( D( LLL+1 ) )*( MU / ( MU+ABS( E( LLL ) ) ) ) |
| 438 |
SMINL = MIN( SMINL, MU ) |
| 439 |
100 CONTINUE |
| 440 |
END IF |
| 441 |
* |
| 442 |
ELSE |
| 443 |
* |
| 444 |
* Run convergence test in backward direction |
| 445 |
* First apply standard test to top of matrix |
| 446 |
* |
| 447 |
IF( ABS( E( LL ) ).LE.ABS( TOL )*ABS( D( LL ) ) .OR. |
| 448 |
$ ( TOL.LT.ZERO .AND. ABS( E( LL ) ).LE.THRESH ) ) THEN |
| 449 |
E( LL ) = ZERO |
| 450 |
GO TO 60 |
| 451 |
END IF |
| 452 |
* |
| 453 |
IF( TOL.GE.ZERO ) THEN |
| 454 |
* |
| 455 |
* If relative accuracy desired, |
| 456 |
* apply convergence criterion backward |
| 457 |
* |
| 458 |
MU = ABS( D( M ) ) |
| 459 |
SMINL = MU |
| 460 |
DO 110 LLL = M - 1, LL, -1 |
| 461 |
IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN |
| 462 |
E( LLL ) = ZERO |
| 463 |
GO TO 60 |
| 464 |
END IF |
| 465 |
MU = ABS( D( LLL ) )*( MU / ( MU+ABS( E( LLL ) ) ) ) |
| 466 |
SMINL = MIN( SMINL, MU ) |
| 467 |
110 CONTINUE |
| 468 |
END IF |
| 469 |
END IF |
| 470 |
OLDLL = LL |
| 471 |
OLDM = M |
| 472 |
* |
| 473 |
* Compute shift. First, test if shifting would ruin relative |
| 474 |
* accuracy, and if so set the shift to zero. |
| 475 |
* |
| 476 |
IF( TOL.GE.ZERO .AND. N*TOL*( SMINL / SMAX ).LE. |
| 477 |
$ MAX( EPS, HNDRTH*TOL ) ) THEN |
| 478 |
* |
| 479 |
* Use a zero shift to avoid loss of relative accuracy |
| 480 |
* |
| 481 |
SHIFT = ZERO |
| 482 |
ELSE |
| 483 |
* |
| 484 |
* Compute the shift from 2-by-2 block at end of matrix |
| 485 |
* |
| 486 |
IF( IDIR.EQ.1 ) THEN |
| 487 |
SLL = ABS( D( LL ) ) |
| 488 |
CALL DLAS2( D( M-1 ), E( M-1 ), D( M ), SHIFT, R ) |
| 489 |
ELSE |
| 490 |
SLL = ABS( D( M ) ) |
| 491 |
CALL DLAS2( D( LL ), E( LL ), D( LL+1 ), SHIFT, R ) |
| 492 |
END IF |
| 493 |
* |
| 494 |
* Test if shift negligible, and if so set to zero |
| 495 |
* |
| 496 |
IF( SLL.GT.ZERO ) THEN |
| 497 |
IF( ( SHIFT / SLL )**2.LT.EPS ) |
| 498 |
$ SHIFT = ZERO |
| 499 |
END IF |
| 500 |
END IF |
| 501 |
* |
| 502 |
* Increment iteration count |
| 503 |
* |
| 504 |
ITER = ITER + M - LL |
| 505 |
* |
| 506 |
* If SHIFT = 0, do simplified QR iteration |
| 507 |
* |
| 508 |
IF( SHIFT.EQ.ZERO ) THEN |
| 509 |
IF( IDIR.EQ.1 ) THEN |
| 510 |
* |
| 511 |
* Chase bulge from top to bottom |
| 512 |
* Save cosines and sines for later singular vector updates |
| 513 |
* |
| 514 |
CS = ONE |
| 515 |
OLDCS = ONE |
| 516 |
DO 120 I = LL, M - 1 |
| 517 |
CALL DLARTG( D( I )*CS, E( I ), CS, SN, R ) |
| 518 |
IF( I.GT.LL ) |
| 519 |
$ E( I-1 ) = OLDSN*R |
| 520 |
CALL DLARTG( OLDCS*R, D( I+1 )*SN, OLDCS, OLDSN, D( I ) ) |
| 521 |
WORK( I-LL+1 ) = CS |
| 522 |
WORK( I-LL+1+NM1 ) = SN |
| 523 |
WORK( I-LL+1+NM12 ) = OLDCS |
| 524 |
WORK( I-LL+1+NM13 ) = OLDSN |
| 525 |
120 CONTINUE |
| 526 |
H = D( M )*CS |
| 527 |
D( M ) = H*OLDCS |
| 528 |
E( M-1 ) = H*OLDSN |
| 529 |
* |
| 530 |
* Update singular vectors |
| 531 |
* |
| 532 |
IF( NCVT.GT.0 ) |
| 533 |
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ), |
| 534 |
$ WORK( N ), VT( LL, 1 ), LDVT ) |
| 535 |
IF( NRU.GT.0 ) |
| 536 |
$ CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1, WORK( NM12+1 ), |
| 537 |
$ WORK( NM13+1 ), U( 1, LL ), LDU ) |
| 538 |
IF( NCC.GT.0 ) |
| 539 |
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC, WORK( NM12+1 ), |
| 540 |
$ WORK( NM13+1 ), C( LL, 1 ), LDC ) |
| 541 |
* |
| 542 |
* Test convergence |
| 543 |
* |
| 544 |
IF( ABS( E( M-1 ) ).LE.THRESH ) |
| 545 |
$ E( M-1 ) = ZERO |
| 546 |
* |
| 547 |
ELSE |
| 548 |
* |
| 549 |
* Chase bulge from bottom to top |
| 550 |
* Save cosines and sines for later singular vector updates |
| 551 |
* |
| 552 |
CS = ONE |
| 553 |
OLDCS = ONE |
| 554 |
DO 130 I = M, LL + 1, -1 |
| 555 |
CALL DLARTG( D( I )*CS, E( I-1 ), CS, SN, R ) |
| 556 |
IF( I.LT.M ) |
| 557 |
$ E( I ) = OLDSN*R |
| 558 |
CALL DLARTG( OLDCS*R, D( I-1 )*SN, OLDCS, OLDSN, D( I ) ) |
| 559 |
WORK( I-LL ) = CS |
| 560 |
WORK( I-LL+NM1 ) = -SN |
| 561 |
WORK( I-LL+NM12 ) = OLDCS |
| 562 |
WORK( I-LL+NM13 ) = -OLDSN |
| 563 |
130 CONTINUE |
| 564 |
H = D( LL )*CS |
| 565 |
D( LL ) = H*OLDCS |
| 566 |
E( LL ) = H*OLDSN |
| 567 |
* |
| 568 |
* Update singular vectors |
| 569 |
* |
| 570 |
IF( NCVT.GT.0 ) |
| 571 |
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT, WORK( NM12+1 ), |
| 572 |
$ WORK( NM13+1 ), VT( LL, 1 ), LDVT ) |
| 573 |
IF( NRU.GT.0 ) |
| 574 |
$ CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ), |
| 575 |
$ WORK( N ), U( 1, LL ), LDU ) |
| 576 |
IF( NCC.GT.0 ) |
| 577 |
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ), |
| 578 |
$ WORK( N ), C( LL, 1 ), LDC ) |
| 579 |
* |
| 580 |
* Test convergence |
| 581 |
* |
| 582 |
IF( ABS( E( LL ) ).LE.THRESH ) |
| 583 |
$ E( LL ) = ZERO |
| 584 |
END IF |
| 585 |
ELSE |
| 586 |
* |
| 587 |
* Use nonzero shift |
| 588 |
* |
| 589 |
IF( IDIR.EQ.1 ) THEN |
| 590 |
* |
| 591 |
* Chase bulge from top to bottom |
| 592 |
* Save cosines and sines for later singular vector updates |
| 593 |
* |
| 594 |
F = ( ABS( D( LL ) )-SHIFT )* |
| 595 |
$ ( SIGN( ONE, D( LL ) )+SHIFT / D( LL ) ) |
| 596 |
G = E( LL ) |
| 597 |
DO 140 I = LL, M - 1 |
| 598 |
CALL DLARTG( F, G, COSR, SINR, R ) |
| 599 |
IF( I.GT.LL ) |
| 600 |
$ E( I-1 ) = R |
| 601 |
F = COSR*D( I ) + SINR*E( I ) |
| 602 |
E( I ) = COSR*E( I ) - SINR*D( I ) |
| 603 |
G = SINR*D( I+1 ) |
| 604 |
D( I+1 ) = COSR*D( I+1 ) |
| 605 |
CALL DLARTG( F, G, COSL, SINL, R ) |
| 606 |
D( I ) = R |
| 607 |
F = COSL*E( I ) + SINL*D( I+1 ) |
| 608 |
D( I+1 ) = COSL*D( I+1 ) - SINL*E( I ) |
| 609 |
IF( I.LT.M-1 ) THEN |
| 610 |
G = SINL*E( I+1 ) |
| 611 |
E( I+1 ) = COSL*E( I+1 ) |
| 612 |
END IF |
| 613 |
WORK( I-LL+1 ) = COSR |
| 614 |
WORK( I-LL+1+NM1 ) = SINR |
| 615 |
WORK( I-LL+1+NM12 ) = COSL |
| 616 |
WORK( I-LL+1+NM13 ) = SINL |
| 617 |
140 CONTINUE |
| 618 |
E( M-1 ) = F |
| 619 |
* |
| 620 |
* Update singular vectors |
| 621 |
* |
| 622 |
IF( NCVT.GT.0 ) |
| 623 |
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ), |
| 624 |
$ WORK( N ), VT( LL, 1 ), LDVT ) |
| 625 |
IF( NRU.GT.0 ) |
| 626 |
$ CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1, WORK( NM12+1 ), |
| 627 |
$ WORK( NM13+1 ), U( 1, LL ), LDU ) |
| 628 |
IF( NCC.GT.0 ) |
| 629 |
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC, WORK( NM12+1 ), |
| 630 |
$ WORK( NM13+1 ), C( LL, 1 ), LDC ) |
| 631 |
* |
| 632 |
* Test convergence |
| 633 |
* |
| 634 |
IF( ABS( E( M-1 ) ).LE.THRESH ) |
| 635 |
$ E( M-1 ) = ZERO |
| 636 |
* |
| 637 |
ELSE |
| 638 |
* |
| 639 |
* Chase bulge from bottom to top |
| 640 |
* Save cosines and sines for later singular vector updates |
| 641 |
* |
| 642 |
F = ( ABS( D( M ) )-SHIFT )*( SIGN( ONE, D( M ) )+SHIFT / |
| 643 |
$ D( M ) ) |
| 644 |
G = E( M-1 ) |
| 645 |
DO 150 I = M, LL + 1, -1 |
| 646 |
CALL DLARTG( F, G, COSR, SINR, R ) |
| 647 |
IF( I.LT.M ) |
| 648 |
$ E( I ) = R |
| 649 |
F = COSR*D( I ) + SINR*E( I-1 ) |
| 650 |
E( I-1 ) = COSR*E( I-1 ) - SINR*D( I ) |
| 651 |
G = SINR*D( I-1 ) |
| 652 |
D( I-1 ) = COSR*D( I-1 ) |
| 653 |
CALL DLARTG( F, G, COSL, SINL, R ) |
| 654 |
D( I ) = R |
| 655 |
F = COSL*E( I-1 ) + SINL*D( I-1 ) |
| 656 |
D( I-1 ) = COSL*D( I-1 ) - SINL*E( I-1 ) |
| 657 |
IF( I.GT.LL+1 ) THEN |
| 658 |
G = SINL*E( I-2 ) |
| 659 |
E( I-2 ) = COSL*E( I-2 ) |
| 660 |
END IF |
| 661 |
WORK( I-LL ) = COSR |
| 662 |
WORK( I-LL+NM1 ) = -SINR |
| 663 |
WORK( I-LL+NM12 ) = COSL |
| 664 |
WORK( I-LL+NM13 ) = -SINL |
| 665 |
150 CONTINUE |
| 666 |
E( LL ) = F |
| 667 |
* |
| 668 |
* Test convergence |
| 669 |
* |
| 670 |
IF( ABS( E( LL ) ).LE.THRESH ) |
| 671 |
$ E( LL ) = ZERO |
| 672 |
* |
| 673 |
* Update singular vectors if desired |
| 674 |
* |
| 675 |
IF( NCVT.GT.0 ) |
| 676 |
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT, WORK( NM12+1 ), |
| 677 |
$ WORK( NM13+1 ), VT( LL, 1 ), LDVT ) |
| 678 |
IF( NRU.GT.0 ) |
| 679 |
$ CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ), |
| 680 |
$ WORK( N ), U( 1, LL ), LDU ) |
| 681 |
IF( NCC.GT.0 ) |
| 682 |
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ), |
| 683 |
$ WORK( N ), C( LL, 1 ), LDC ) |
| 684 |
END IF |
| 685 |
END IF |
| 686 |
* |
| 687 |
* QR iteration finished, go back and check convergence |
| 688 |
* |
| 689 |
GO TO 60 |
| 690 |
* |
| 691 |
* All singular values converged, so make them positive |
| 692 |
* |
| 693 |
160 CONTINUE |
| 694 |
DO 170 I = 1, N |
| 695 |
IF( D( I ).LT.ZERO ) THEN |
| 696 |
D( I ) = -D( I ) |
| 697 |
* |
| 698 |
* Change sign of singular vectors, if desired |
| 699 |
* |
| 700 |
IF( NCVT.GT.0 ) |
| 701 |
$ CALL DSCAL( NCVT, NEGONE, VT( I, 1 ), LDVT ) |
| 702 |
END IF |
| 703 |
170 CONTINUE |
| 704 |
* |
| 705 |
* Sort the singular values into decreasing order (insertion sort on |
| 706 |
* singular values, but only one transposition per singular vector) |
| 707 |
* |
| 708 |
DO 190 I = 1, N - 1 |
| 709 |
* |
| 710 |
* Scan for smallest D(I) |
| 711 |
* |
| 712 |
ISUB = 1 |
| 713 |
SMIN = D( 1 ) |
| 714 |
DO 180 J = 2, N + 1 - I |
| 715 |
IF( D( J ).LE.SMIN ) THEN |
| 716 |
ISUB = J |
| 717 |
SMIN = D( J ) |
| 718 |
END IF |
| 719 |
180 CONTINUE |
| 720 |
IF( ISUB.NE.N+1-I ) THEN |
| 721 |
* |
| 722 |
* Swap singular values and vectors |
| 723 |
* |
| 724 |
D( ISUB ) = D( N+1-I ) |
| 725 |
D( N+1-I ) = SMIN |
| 726 |
IF( NCVT.GT.0 ) |
| 727 |
$ CALL DSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( N+1-I, 1 ), |
| 728 |
$ LDVT ) |
| 729 |
IF( NRU.GT.0 ) |
| 730 |
$ CALL DSWAP( NRU, U( 1, ISUB ), 1, U( 1, N+1-I ), 1 ) |
| 731 |
IF( NCC.GT.0 ) |
| 732 |
$ CALL DSWAP( NCC, C( ISUB, 1 ), LDC, C( N+1-I, 1 ), LDC ) |
| 733 |
END IF |
| 734 |
190 CONTINUE |
| 735 |
GO TO 220 |
| 736 |
* |
| 737 |
* Maximum number of iterations exceeded, failure to converge |
| 738 |
* |
| 739 |
200 CONTINUE |
| 740 |
INFO = 0 |
| 741 |
DO 210 I = 1, N - 1 |
| 742 |
IF( E( I ).NE.ZERO ) |
| 743 |
$ INFO = INFO + 1 |
| 744 |
210 CONTINUE |
| 745 |
220 CONTINUE |
| 746 |
RETURN |
| 747 |
* |
| 748 |
* End of DBDSQR |
| 749 |
* |
| 750 |
END |