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SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
$ LDQ, PT, LDPT, C, LDC, WORK, INFO )
*
* -- LAPACK routine (version 3.3.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* -- April 2011 --
*
* .. 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( * )
* ..
*
* Purpose
* =======
*
* 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.
*
* Arguments
* =========
*
* VECT (input) 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.
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* NCC (input) INTEGER
* The number of columns of the matrix C. NCC >= 0.
*
* KL (input) INTEGER
* The number of subdiagonals of the matrix A. KL >= 0.
*
* KU (input) INTEGER
* The number of superdiagonals of the matrix A. KU >= 0.
*
* AB (input/output) 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.
*
* LDAB (input) INTEGER
* The leading dimension of the array A. LDAB >= KL+KU+1.
*
* D (output) DOUBLE PRECISION array, dimension (min(M,N))
* The diagonal elements of the bidiagonal matrix B.
*
* E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
* The superdiagonal elements of the bidiagonal matrix B.
*
* Q (output) 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.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q.
* LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
*
* PT (output) 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.
*
* LDPT (input) INTEGER
* The leading dimension of the array PT.
* LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
*
* C (input/output) 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.
*
* LDC (input) INTEGER
* The leading dimension of the array C.
* LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
*
* WORK (workspace) DOUBLE PRECISION array, dimension (2*max(M,N))
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* =====================================================================
*
* .. 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
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