SUBROUTINE SPBTRF( UPLO, N, KD, AB, LDAB, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
REAL AB( LDAB, * )
* ..
*
* Purpose
* =======
*
* SPBTRF computes the Cholesky factorization of a real symmetric
* positive definite band matrix A.
*
* The factorization has the form
* A = U**T * U, if UPLO = 'U', or
* A = L * L**T, if UPLO = 'L',
* where U is an upper triangular matrix and L is lower triangular.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* AB (input/output) REAL array, dimension (LDAB,N)
* On entry, the upper or lower triangle of the symmetric band
* matrix A, stored in the first KD+1 rows of the array. The
* j-th column of A is stored in the j-th column of the array AB
* as follows:
* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*
* On exit, if INFO = 0, the triangular factor U or L from the
* Cholesky factorization A = U**T*U or A = L*L**T of the band
* matrix A, in the same storage format as A.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, the leading minor of order i is not
* positive definite, and the factorization could not be
* completed.
*
* Further Details
* ===============
*
* The band storage scheme is illustrated by the following example, when
* N = 6, KD = 2, and UPLO = 'U':
*
* On entry: On exit:
*
* * * a13 a24 a35 a46 * * u13 u24 u35 u46
* * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
* a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*
* Similarly, if UPLO = 'L' the format of A is as follows:
*
* On entry: On exit:
*
* a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
* a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
* a31 a42 a53 a64 * * l31 l42 l53 l64 * *
*
* Array elements marked * are not used by the routine.
*
* Contributed by
* Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
INTEGER NBMAX, LDWORK
PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 )
* ..
* .. Local Scalars ..
INTEGER I, I2, I3, IB, II, J, JJ, NB
* ..
* .. Local Arrays ..
REAL WORK( LDWORK, NBMAX )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL SGEMM, SPBTF2, SPOTF2, SSYRK, STRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
$ ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SPBTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment
*
NB = ILAENV( 1, 'SPBTRF', UPLO, N, KD, -1, -1 )
*
* The block size must not exceed the semi-bandwidth KD, and must not
* exceed the limit set by the size of the local array WORK.
*
NB = MIN( NB, NBMAX )
*
IF( NB.LE.1 .OR. NB.GT.KD ) THEN
*
* Use unblocked code
*
CALL SPBTF2( UPLO, N, KD, AB, LDAB, INFO )
ELSE
*
* Use blocked code
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Compute the Cholesky factorization of a symmetric band
* matrix, given the upper triangle of the matrix in band
* storage.
*
* Zero the upper triangle of the work array.
*
DO 20 J = 1, NB
DO 10 I = 1, J - 1
WORK( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
*
* Process the band matrix one diagonal block at a time.
*
DO 70 I = 1, N, NB
IB = MIN( NB, N-I+1 )
*
* Factorize the diagonal block
*
CALL SPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
IF( II.NE.0 ) THEN
INFO = I + II - 1
GO TO 150
END IF
IF( I+IB.LE.N ) THEN
*
* Update the relevant part of the trailing submatrix.
* If A11 denotes the diagonal block which has just been
* factorized, then we need to update the remaining
* blocks in the diagram:
*
* A11 A12 A13
* A22 A23
* A33
*
* The numbers of rows and columns in the partitioning
* are IB, I2, I3 respectively. The blocks A12, A22 and
* A23 are empty if IB = KD. The upper triangle of A13
* lies outside the band.
*
I2 = MIN( KD-IB, N-I-IB+1 )
I3 = MIN( IB, N-I-KD+1 )
*
IF( I2.GT.0 ) THEN
*
* Update A12
*
CALL STRSM( 'Left', 'Upper', 'Transpose',
$ 'Non-unit', IB, I2, ONE, AB( KD+1, I ),
$ LDAB-1, AB( KD+1-IB, I+IB ), LDAB-1 )
*
* Update A22
*
CALL SSYRK( 'Upper', 'Transpose', I2, IB, -ONE,
$ AB( KD+1-IB, I+IB ), LDAB-1, ONE,
$ AB( KD+1, I+IB ), LDAB-1 )
END IF
*
IF( I3.GT.0 ) THEN
*
* Copy the lower triangle of A13 into the work array.
*
DO 40 JJ = 1, I3
DO 30 II = JJ, IB
WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 )
30 CONTINUE
40 CONTINUE
*
* Update A13 (in the work array).
*
CALL STRSM( 'Left', 'Upper', 'Transpose',
$ 'Non-unit', IB, I3, ONE, AB( KD+1, I ),
$ LDAB-1, WORK, LDWORK )
*
* Update A23
*
IF( I2.GT.0 )
$ CALL SGEMM( 'Transpose', 'No Transpose', I2, I3,
$ IB, -ONE, AB( KD+1-IB, I+IB ),
$ LDAB-1, WORK, LDWORK, ONE,
$ AB( 1+IB, I+KD ), LDAB-1 )
*
* Update A33
*
CALL SSYRK( 'Upper', 'Transpose', I3, IB, -ONE,
$ WORK, LDWORK, ONE, AB( KD+1, I+KD ),
$ LDAB-1 )
*
* Copy the lower triangle of A13 back into place.
*
DO 60 JJ = 1, I3
DO 50 II = JJ, IB
AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
50 CONTINUE
60 CONTINUE
END IF
END IF
70 CONTINUE
ELSE
*
* Compute the Cholesky factorization of a symmetric band
* matrix, given the lower triangle of the matrix in band
* storage.
*
* Zero the lower triangle of the work array.
*
DO 90 J = 1, NB
DO 80 I = J + 1, NB
WORK( I, J ) = ZERO
80 CONTINUE
90 CONTINUE
*
* Process the band matrix one diagonal block at a time.
*
DO 140 I = 1, N, NB
IB = MIN( NB, N-I+1 )
*
* Factorize the diagonal block
*
CALL SPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
IF( II.NE.0 ) THEN
INFO = I + II - 1
GO TO 150
END IF
IF( I+IB.LE.N ) THEN
*
* Update the relevant part of the trailing submatrix.
* If A11 denotes the diagonal block which has just been
* factorized, then we need to update the remaining
* blocks in the diagram:
*
* A11
* A21 A22
* A31 A32 A33
*
* The numbers of rows and columns in the partitioning
* are IB, I2, I3 respectively. The blocks A21, A22 and
* A32 are empty if IB = KD. The lower triangle of A31
* lies outside the band.
*
I2 = MIN( KD-IB, N-I-IB+1 )
I3 = MIN( IB, N-I-KD+1 )
*
IF( I2.GT.0 ) THEN
*
* Update A21
*
CALL STRSM( 'Right', 'Lower', 'Transpose',
$ 'Non-unit', I2, IB, ONE, AB( 1, I ),
$ LDAB-1, AB( 1+IB, I ), LDAB-1 )
*
* Update A22
*
CALL SSYRK( 'Lower', 'No Transpose', I2, IB, -ONE,
$ AB( 1+IB, I ), LDAB-1, ONE,
$ AB( 1, I+IB ), LDAB-1 )
END IF
*
IF( I3.GT.0 ) THEN
*
* Copy the upper triangle of A31 into the work array.
*
DO 110 JJ = 1, IB
DO 100 II = 1, MIN( JJ, I3 )
WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
100 CONTINUE
110 CONTINUE
*
* Update A31 (in the work array).
*
CALL STRSM( 'Right', 'Lower', 'Transpose',
$ 'Non-unit', I3, IB, ONE, AB( 1, I ),
$ LDAB-1, WORK, LDWORK )
*
* Update A32
*
IF( I2.GT.0 )
$ CALL SGEMM( 'No transpose', 'Transpose', I3, I2,
$ IB, -ONE, WORK, LDWORK,
$ AB( 1+IB, I ), LDAB-1, ONE,
$ AB( 1+KD-IB, I+IB ), LDAB-1 )
*
* Update A33
*
CALL SSYRK( 'Lower', 'No Transpose', I3, IB, -ONE,
$ WORK, LDWORK, ONE, AB( 1, I+KD ),
$ LDAB-1 )
*
* Copy the upper triangle of A31 back into place.
*
DO 130 JJ = 1, IB
DO 120 II = 1, MIN( JJ, I3 )
AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
120 CONTINUE
130 CONTINUE
END IF
END IF
140 CONTINUE
END IF
END IF
RETURN
*
150 CONTINUE
RETURN
*
* End of SPBTRF
*
END