**Purpose**

To compute all, or part, of the singular value decomposition of a real upper triangular matrix. The N-by-N upper triangular matrix A is factored as A = Q*S*P', where Q and P are N-by-N orthogonal matrices and S is an N-by-N diagonal matrix with non-negative diagonal elements, SV(1), SV(2), ..., SV(N), ordered such that SV(1) >= SV(2) >= ... >= SV(N) >= 0. The columns of Q are the left singular vectors of A, the diagonal elements of S are the singular values of A and the columns of P are the right singular vectors of A. Either or both of Q and P' may be requested. When P' is computed, it is returned in A.

SUBROUTINE MB03UD( JOBQ, JOBP, N, A, LDA, Q, LDQ, SV, DWORK, $ LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER JOBP, JOBQ INTEGER INFO, LDA, LDQ, LDWORK, N C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), DWORK(*), Q(LDQ,*), SV(*)

**Mode Parameters**

JOBQ CHARACTER*1 Specifies whether the user wishes to compute the matrix Q of left singular vectors as follows: = 'V': Left singular vectors are computed; = 'N': No left singular vectors are computed. JOBP CHARACTER*1 Specifies whether the user wishes to compute the matrix P' of right singular vectors as follows: = 'V': Right singular vectors are computed; = 'N': No right singular vectors are computed.

N (input) INTEGER The order of the matrix A. N >= 0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N upper triangular part of this array must contain the upper triangular matrix A. On exit, if JOBP = 'V', the leading N-by-N part of this array contains the N-by-N orthogonal matrix P'; otherwise the N-by-N upper triangular part of A is used as internal workspace. The strictly lower triangular part of A is set internally to zero before the reduction to bidiagonal form is performed. LDA INTEGER The leading dimension of array A. LDA >= MAX(1,N). Q (output) DOUBLE PRECISION array, dimension (LDQ,N) If JOBQ = 'V', the leading N-by-N part of this array contains the orthogonal matrix Q. If JOBQ = 'N', Q is not referenced. LDQ INTEGER The leading dimension of array Q. LDQ >= 1, and when JOBQ = 'V', LDQ >= MAX(1,N). SV (output) DOUBLE PRECISION array, dimension (N) The N singular values of the matrix A, sorted in descending order.

DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK; if INFO > 0, DWORK(2:N) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in SV (not necessarily sorted). B satisfies A = Q*B*P', so it has the same singular values as A, and singular vectors related by Q and P'. LDWORK INTEGER The length of the array DWORK. LDWORK >= MAX(1,5*N). For optimum performance LDWORK should be larger. If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the DWORK array, returns this value as the first entry of the DWORK array, and no error message related to LDWORK is issued by XERBLA.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; > 0: the QR algorithm has failed to converge. In this case INFO specifies how many superdiagonals did not converge (see the description of DWORK). This failure is not likely to occur.

The routine reduces A to bidiagonal form by means of elementary reflectors and then uses the QR algorithm on the bidiagonal form.

None

**Program Text**

* MB03UD EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2017 NICONET e.V. * * .. Parameters .. INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER NMAX PARAMETER ( NMAX = 10 ) INTEGER LDA, LDQ PARAMETER ( LDA = NMAX, LDQ = NMAX ) INTEGER LDWORK PARAMETER ( LDWORK = MAX( 1, 5*NMAX ) ) * .. Local Scalars .. CHARACTER*1 JOBQ, JOBP INTEGER I, INFO, J, N * .. Local Arrays .. DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), Q(LDQ,NMAX), $ SV(NMAX) * .. External Functions .. LOGICAL LSAME * .. External Subroutines .. EXTERNAL MB03UD * .. Intrinsic Functions .. INTRINSIC MAX * .. Executable Statements .. * WRITE ( NOUT, FMT = 99999 ) * Skip the heading in the data file and read the data. READ ( NIN, FMT = '()' ) READ ( NIN, FMT = * ) N, JOBQ, JOBP IF ( N.LT.0 .OR. N.GT.NMAX ) THEN WRITE ( NOUT, FMT = 99993 ) N ELSE READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N ) * Compute the singular values and vectors. CALL MB03UD( JOBQ, JOBP, N, A, LDA, Q, LDQ, SV, DWORK, $ LDWORK, INFO ) IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE WRITE ( NOUT, FMT = 99997 ) WRITE ( NOUT, FMT = 99995 ) ( SV(I), I = 1,N ) IF ( LSAME( JOBP, 'V' ) ) THEN WRITE ( NOUT, FMT = 99996 ) DO 10 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N ) 10 CONTINUE END IF IF ( LSAME( JOBQ, 'V' ) ) THEN WRITE ( NOUT, FMT = 99994 ) DO 20 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N ) 20 CONTINUE END IF END IF END IF * STOP * 99999 FORMAT (' MB03UD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from MB03UD = ',I2) 99997 FORMAT (' Singular values are ',I5) 99996 FORMAT (/' The transpose of the right singular vectors matrix is ' $ ) 99995 FORMAT (8X,20(1X,F8.4)) 99994 FORMAT (/' The left singular vectors matrix is ') 99993 FORMAT (/' N is out of range.',/' N = ',I5) END

MB03UD EXAMPLE PROGRAM DATA 4 V V -1.0 37.0 -12.0 -12.0 0.0 -10.0 0.0 4.0 0.0 0.0 7.0 -6.0 0.0 0.0 0.0 -9.0

MB03UD EXAMPLE PROGRAM RESULTS Singular values are 42.0909 11.7764 5.4420 0.2336 The transpose of the right singular vectors matrix is 0.0230 -0.9084 0.2759 0.3132 0.0075 -0.1272 0.5312 -0.8376 0.0092 0.3978 0.8009 0.4476 0.9997 0.0182 -0.0177 -0.0050 The left singular vectors matrix is -0.9671 -0.0882 -0.0501 -0.2335 0.2456 -0.1765 -0.4020 -0.8643 0.0012 0.7425 0.5367 -0.4008 -0.0670 0.6401 -0.7402 0.1945