*> \brief \b DGBBRD * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DGBBRD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q, * LDQ, PT, LDPT, C, LDC, WORK, INFO ) * * .. Scalar Arguments .. * CHARACTER VECT * INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC * .. * .. Array Arguments .. * DOUBLE PRECISION AB( LDAB, * ), C( LDC, * ), D( * ), E( * ), * $ PT( LDPT, * ), Q( LDQ, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DGBBRD reduces a real general m-by-n band matrix A to upper *> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. *> *> The routine computes B, and optionally forms Q or P**T, or computes *> Q**T*C for a given matrix C. *> \endverbatim * * Arguments: * ========== * *> \param[in] VECT *> \verbatim *> VECT is CHARACTER*1 *> Specifies whether or not the matrices Q and P**T are to be *> formed. *> = 'N': do not form Q or P**T; *> = 'Q': form Q only; *> = 'P': form P**T only; *> = 'B': form both. *> \endverbatim *> *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] NCC *> \verbatim *> NCC is INTEGER *> The number of columns of the matrix C. NCC >= 0. *> \endverbatim *> *> \param[in] KL *> \verbatim *> KL is INTEGER *> The number of subdiagonals of the matrix A. KL >= 0. *> \endverbatim *> *> \param[in] KU *> \verbatim *> KU is INTEGER *> The number of superdiagonals of the matrix A. KU >= 0. *> \endverbatim *> *> \param[in,out] AB *> \verbatim *> AB is DOUBLE PRECISION array, dimension (LDAB,N) *> On entry, the m-by-n band matrix A, stored in rows 1 to *> KL+KU+1. The j-th column of A is stored in the j-th column of *> the array AB as follows: *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl). *> On exit, A is overwritten by values generated during the *> reduction. *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array A. LDAB >= KL+KU+1. *> \endverbatim *> *> \param[out] D *> \verbatim *> D is DOUBLE PRECISION array, dimension (min(M,N)) *> The diagonal elements of the bidiagonal matrix B. *> \endverbatim *> *> \param[out] E *> \verbatim *> E is DOUBLE PRECISION array, dimension (min(M,N)-1) *> The superdiagonal elements of the bidiagonal matrix B. *> \endverbatim *> *> \param[out] Q *> \verbatim *> Q is DOUBLE PRECISION array, dimension (LDQ,M) *> If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q. *> If VECT = 'N' or 'P', the array Q is not referenced. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. *> LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise. *> \endverbatim *> *> \param[out] PT *> \verbatim *> PT is DOUBLE PRECISION array, dimension (LDPT,N) *> If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'. *> If VECT = 'N' or 'Q', the array PT is not referenced. *> \endverbatim *> *> \param[in] LDPT *> \verbatim *> LDPT is INTEGER *> The leading dimension of the array PT. *> LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise. *> \endverbatim *> *> \param[in,out] C *> \verbatim *> C is DOUBLE PRECISION array, dimension (LDC,NCC) *> On entry, an m-by-ncc matrix C. *> On exit, C is overwritten by Q**T*C. *> C is not referenced if NCC = 0. *> \endverbatim *> *> \param[in] LDC *> \verbatim *> LDC is INTEGER *> The leading dimension of the array C. *> LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (2*max(M,N)) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit. *> < 0: if INFO = -i, the i-th argument had an illegal value. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup doubleGBcomputational * * ===================================================================== SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q, $ LDQ, PT, LDPT, C, LDC, WORK, INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER VECT INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC * .. * .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), C( LDC, * ), D( * ), E( * ), $ PT( LDPT, * ), Q( LDQ, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL WANTB, WANTC, WANTPT, WANTQ INTEGER I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1, $ KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT DOUBLE PRECISION RA, RB, RC, RS * .. * .. External Subroutines .. EXTERNAL DLARGV, DLARTG, DLARTV, DLASET, DROT, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. Executable Statements .. * * Test the input parameters * WANTB = LSAME( VECT, 'B' ) WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB WANTPT = LSAME( VECT, 'P' ) .OR. WANTB WANTC = NCC.GT.0 KLU1 = KL + KU + 1 INFO = 0 IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) ) $ THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NCC.LT.0 ) THEN INFO = -4 ELSE IF( KL.LT.0 ) THEN INFO = -5 ELSE IF( KU.LT.0 ) THEN INFO = -6 ELSE IF( LDAB.LT.KLU1 ) THEN INFO = -8 ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN INFO = -12 ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN INFO = -14 ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN INFO = -16 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBBRD', -INFO ) RETURN END IF * * Initialize Q and P**T to the unit matrix, if needed * IF( WANTQ ) $ CALL DLASET( 'Full', M, M, ZERO, ONE, Q, LDQ ) IF( WANTPT ) $ CALL DLASET( 'Full', N, N, ZERO, ONE, PT, LDPT ) * * Quick return if possible. * IF( M.EQ.0 .OR. N.EQ.0 ) $ RETURN * MINMN = MIN( M, N ) * IF( KL+KU.GT.1 ) THEN * * Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce * first to lower bidiagonal form and then transform to upper * bidiagonal * IF( KU.GT.0 ) THEN ML0 = 1 MU0 = 2 ELSE ML0 = 2 MU0 = 1 END IF * * Wherever possible, plane rotations are generated and applied in * vector operations of length NR over the index set J1:J2:KLU1. * * The sines of the plane rotations are stored in WORK(1:max(m,n)) * and the cosines in WORK(max(m,n)+1:2*max(m,n)). * MN = MAX( M, N ) KLM = MIN( M-1, KL ) KUN = MIN( N-1, KU ) KB = KLM + KUN KB1 = KB + 1 INCA = KB1*LDAB NR = 0 J1 = KLM + 2 J2 = 1 - KUN * DO 90 I = 1, MINMN * * Reduce i-th column and i-th row of matrix to bidiagonal form * ML = KLM + 1 MU = KUN + 1 DO 80 KK = 1, KB J1 = J1 + KB J2 = J2 + KB * * generate plane rotations to annihilate nonzero elements * which have been created below the band * IF( NR.GT.0 ) $ CALL DLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA, $ WORK( J1 ), KB1, WORK( MN+J1 ), KB1 ) * * apply plane rotations from the left * DO 10 L = 1, KB IF( J2-KLM+L-1.GT.N ) THEN NRT = NR - 1 ELSE NRT = NR END IF IF( NRT.GT.0 ) $ CALL DLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA, $ AB( KLU1-L+1, J1-KLM+L-1 ), INCA, $ WORK( MN+J1 ), WORK( J1 ), KB1 ) 10 CONTINUE * IF( ML.GT.ML0 ) THEN IF( ML.LE.M-I+1 ) THEN * * generate plane rotation to annihilate a(i+ml-1,i) * within the band, and apply rotation from the left * CALL DLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ), $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ), $ RA ) AB( KU+ML-1, I ) = RA IF( I.LT.N ) $ CALL DROT( MIN( KU+ML-2, N-I ), $ AB( KU+ML-2, I+1 ), LDAB-1, $ AB( KU+ML-1, I+1 ), LDAB-1, $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ) ) END IF NR = NR + 1 J1 = J1 - KB1 END IF * IF( WANTQ ) THEN * * accumulate product of plane rotations in Q * DO 20 J = J1, J2, KB1 CALL DROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1, $ WORK( MN+J ), WORK( J ) ) 20 CONTINUE END IF * IF( WANTC ) THEN * * apply plane rotations to C * DO 30 J = J1, J2, KB1 CALL DROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC, $ WORK( MN+J ), WORK( J ) ) 30 CONTINUE END IF * IF( J2+KUN.GT.N ) THEN * * adjust J2 to keep within the bounds of the matrix * NR = NR - 1 J2 = J2 - KB1 END IF * DO 40 J = J1, J2, KB1 * * create nonzero element a(j-1,j+ku) above the band * and store it in WORK(n+1:2*n) * WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN ) AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN ) 40 CONTINUE * * generate plane rotations to annihilate nonzero elements * which have been generated above the band * IF( NR.GT.0 ) $ CALL DLARGV( NR, AB( 1, J1+KUN-1 ), INCA, $ WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ), $ KB1 ) * * apply plane rotations from the right * DO 50 L = 1, KB IF( J2+L-1.GT.M ) THEN NRT = NR - 1 ELSE NRT = NR END IF IF( NRT.GT.0 ) $ CALL DLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA, $ AB( L, J1+KUN ), INCA, $ WORK( MN+J1+KUN ), WORK( J1+KUN ), $ KB1 ) 50 CONTINUE * IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN IF( MU.LE.N-I+1 ) THEN * * generate plane rotation to annihilate a(i,i+mu-1) * within the band, and apply rotation from the right * CALL DLARTG( AB( KU-MU+3, I+MU-2 ), $ AB( KU-MU+2, I+MU-1 ), $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ), $ RA ) AB( KU-MU+3, I+MU-2 ) = RA CALL DROT( MIN( KL+MU-2, M-I ), $ AB( KU-MU+4, I+MU-2 ), 1, $ AB( KU-MU+3, I+MU-1 ), 1, $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ) ) END IF NR = NR + 1 J1 = J1 - KB1 END IF * IF( WANTPT ) THEN * * accumulate product of plane rotations in P**T * DO 60 J = J1, J2, KB1 CALL DROT( N, PT( J+KUN-1, 1 ), LDPT, $ PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ), $ WORK( J+KUN ) ) 60 CONTINUE END IF * IF( J2+KB.GT.M ) THEN * * adjust J2 to keep within the bounds of the matrix * NR = NR - 1 J2 = J2 - KB1 END IF * DO 70 J = J1, J2, KB1 * * create nonzero element a(j+kl+ku,j+ku-1) below the * band and store it in WORK(1:n) * WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN ) AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN ) 70 CONTINUE * IF( ML.GT.ML0 ) THEN ML = ML - 1 ELSE MU = MU - 1 END IF 80 CONTINUE 90 CONTINUE END IF * IF( KU.EQ.0 .AND. KL.GT.0 ) THEN * * A has been reduced to lower bidiagonal form * * Transform lower bidiagonal form to upper bidiagonal by applying * plane rotations from the left, storing diagonal elements in D * and off-diagonal elements in E * DO 100 I = 1, MIN( M-1, N ) CALL DLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA ) D( I ) = RA IF( I.LT.N ) THEN E( I ) = RS*AB( 1, I+1 ) AB( 1, I+1 ) = RC*AB( 1, I+1 ) END IF IF( WANTQ ) $ CALL DROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS ) IF( WANTC ) $ CALL DROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC, $ RS ) 100 CONTINUE IF( M.LE.N ) $ D( M ) = AB( 1, M ) ELSE IF( KU.GT.0 ) THEN * * A has been reduced to upper bidiagonal form * IF( M.LT.N ) THEN * * Annihilate a(m,m+1) by applying plane rotations from the * right, storing diagonal elements in D and off-diagonal * elements in E * RB = AB( KU, M+1 ) DO 110 I = M, 1, -1 CALL DLARTG( AB( KU+1, I ), RB, RC, RS, RA ) D( I ) = RA IF( I.GT.1 ) THEN RB = -RS*AB( KU, I ) E( I-1 ) = RC*AB( KU, I ) END IF IF( WANTPT ) $ CALL DROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT, $ RC, RS ) 110 CONTINUE ELSE * * Copy off-diagonal elements to E and diagonal elements to D * DO 120 I = 1, MINMN - 1 E( I ) = AB( KU, I+1 ) 120 CONTINUE DO 130 I = 1, MINMN D( I ) = AB( KU+1, I ) 130 CONTINUE END IF ELSE * * A is diagonal. Set elements of E to zero and copy diagonal * elements to D. * DO 140 I = 1, MINMN - 1 E( I ) = ZERO 140 CONTINUE DO 150 I = 1, MINMN D( I ) = AB( 1, I ) 150 CONTINUE END IF RETURN * * End of DGBBRD * END