642 lines
20 KiB
Fortran
642 lines
20 KiB
Fortran
*> \brief \b SSBTRD
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download SSBTRD + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssbtrd.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssbtrd.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssbtrd.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE SSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ,
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* WORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER UPLO, VECT
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* INTEGER INFO, KD, LDAB, LDQ, N
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* ..
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* .. Array Arguments ..
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* REAL AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ),
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* $ WORK( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> SSBTRD reduces a real symmetric band matrix A to symmetric
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*> tridiagonal form T by an orthogonal similarity transformation:
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*> Q**T * A * Q = T.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] VECT
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*> \verbatim
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*> VECT is CHARACTER*1
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*> = 'N': do not form Q;
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*> = 'V': form Q;
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*> = 'U': update a matrix X, by forming X*Q.
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*> \endverbatim
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*>
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*> \param[in] UPLO
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*> \verbatim
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*> UPLO is CHARACTER*1
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*> = 'U': Upper triangle of A is stored;
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*> = 'L': Lower triangle of A is stored.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] KD
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*> \verbatim
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*> KD is INTEGER
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*> The number of superdiagonals of the matrix A if UPLO = 'U',
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*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] AB
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*> \verbatim
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*> AB is REAL array, dimension (LDAB,N)
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*> On entry, the upper or lower triangle of the symmetric band
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*> matrix A, stored in the first KD+1 rows of the array. The
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*> j-th column of A is stored in the j-th column of the array AB
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*> as follows:
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*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
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*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
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*> On exit, the diagonal elements of AB are overwritten by the
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*> diagonal elements of the tridiagonal matrix T; if KD > 0, the
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*> elements on the first superdiagonal (if UPLO = 'U') or the
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*> first subdiagonal (if UPLO = 'L') are overwritten by the
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*> off-diagonal elements of T; the rest of AB is overwritten by
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*> values generated during the reduction.
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*> \endverbatim
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*>
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*> \param[in] LDAB
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*> \verbatim
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*> LDAB is INTEGER
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*> The leading dimension of the array AB. LDAB >= KD+1.
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*> \endverbatim
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*>
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*> \param[out] D
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*> \verbatim
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*> D is REAL array, dimension (N)
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*> The diagonal elements of the tridiagonal matrix T.
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*> \endverbatim
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*>
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*> \param[out] E
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*> \verbatim
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*> E is REAL array, dimension (N-1)
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*> The off-diagonal elements of the tridiagonal matrix T:
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*> E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
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*> \endverbatim
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*>
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*> \param[in,out] Q
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*> \verbatim
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*> Q is REAL array, dimension (LDQ,N)
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*> On entry, if VECT = 'U', then Q must contain an N-by-N
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*> matrix X; if VECT = 'N' or 'V', then Q need not be set.
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*>
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*> On exit:
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*> if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
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*> if VECT = 'U', Q contains the product X*Q;
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*> if VECT = 'N', the array Q is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDQ
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*> \verbatim
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*> LDQ is INTEGER
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*> The leading dimension of the array Q.
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*> LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is REAL array, dimension (N)
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is 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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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date November 2011
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*
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*> \ingroup realOTHERcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> Modified by Linda Kaufman, Bell Labs.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE SSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ,
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$ WORK, INFO )
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*
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* -- LAPACK computational routine (version 3.4.0) --
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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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* November 2011
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*
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* .. Scalar Arguments ..
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CHARACTER UPLO, VECT
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INTEGER INFO, KD, LDAB, LDQ, N
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* ..
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* .. Array Arguments ..
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REAL AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ),
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$ WORK( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL INITQ, UPPER, WANTQ
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INTEGER I, I2, IBL, INCA, INCX, IQAEND, IQB, IQEND, J,
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$ J1, J1END, J1INC, J2, JEND, JIN, JINC, K, KD1,
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$ KDM1, KDN, L, LAST, LEND, NQ, NR, NRT
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REAL TEMP
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* ..
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* .. External Subroutines ..
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EXTERNAL SLAR2V, SLARGV, SLARTG, SLARTV, SLASET, SROT,
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$ XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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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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INITQ = LSAME( VECT, 'V' )
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WANTQ = INITQ .OR. LSAME( VECT, 'U' )
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UPPER = LSAME( UPLO, 'U' )
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KD1 = KD + 1
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KDM1 = KD - 1
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INCX = LDAB - 1
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IQEND = 1
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*
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INFO = 0
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IF( .NOT.WANTQ .AND. .NOT.LSAME( VECT, 'N' ) ) THEN
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INFO = -1
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ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( KD.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDAB.LT.KD1 ) THEN
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INFO = -6
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ELSE IF( LDQ.LT.MAX( 1, N ) .AND. WANTQ ) THEN
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INFO = -10
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SSBTRD', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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*
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* Initialize Q to the unit matrix, if needed
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*
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IF( INITQ )
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$ CALL SLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
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*
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* Wherever possible, plane rotations are generated and applied in
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* vector operations of length NR over the index set J1:J2:KD1.
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*
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* The cosines and sines of the plane rotations are stored in the
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* arrays D and WORK.
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*
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INCA = KD1*LDAB
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KDN = MIN( N-1, KD )
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IF( UPPER ) THEN
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*
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IF( KD.GT.1 ) THEN
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*
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* Reduce to tridiagonal form, working with upper triangle
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*
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NR = 0
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J1 = KDN + 2
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J2 = 1
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*
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DO 90 I = 1, N - 2
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*
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* Reduce i-th row of matrix to tridiagonal form
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*
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DO 80 K = KDN + 1, 2, -1
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J1 = J1 + KDN
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J2 = J2 + KDN
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*
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IF( NR.GT.0 ) THEN
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*
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* generate plane rotations to annihilate nonzero
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* elements which have been created outside the band
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*
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CALL SLARGV( NR, AB( 1, J1-1 ), INCA, WORK( J1 ),
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$ KD1, D( J1 ), KD1 )
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*
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* apply rotations from the right
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*
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*
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* Dependent on the the number of diagonals either
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* SLARTV or SROT is used
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*
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IF( NR.GE.2*KD-1 ) THEN
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DO 10 L = 1, KD - 1
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CALL SLARTV( NR, AB( L+1, J1-1 ), INCA,
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$ AB( L, J1 ), INCA, D( J1 ),
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$ WORK( J1 ), KD1 )
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10 CONTINUE
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*
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ELSE
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JEND = J1 + ( NR-1 )*KD1
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DO 20 JINC = J1, JEND, KD1
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CALL SROT( KDM1, AB( 2, JINC-1 ), 1,
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$ AB( 1, JINC ), 1, D( JINC ),
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$ WORK( JINC ) )
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20 CONTINUE
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END IF
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END IF
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*
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*
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IF( K.GT.2 ) THEN
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IF( K.LE.N-I+1 ) THEN
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*
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* generate plane rotation to annihilate a(i,i+k-1)
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* within the band
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*
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CALL SLARTG( AB( KD-K+3, I+K-2 ),
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$ AB( KD-K+2, I+K-1 ), D( I+K-1 ),
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$ WORK( I+K-1 ), TEMP )
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AB( KD-K+3, I+K-2 ) = TEMP
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*
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* apply rotation from the right
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*
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CALL SROT( K-3, AB( KD-K+4, I+K-2 ), 1,
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$ AB( KD-K+3, I+K-1 ), 1, D( I+K-1 ),
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$ WORK( I+K-1 ) )
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END IF
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NR = NR + 1
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J1 = J1 - KDN - 1
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END IF
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*
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* apply plane rotations from both sides to diagonal
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* blocks
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*
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IF( NR.GT.0 )
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$ CALL SLAR2V( NR, AB( KD1, J1-1 ), AB( KD1, J1 ),
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$ AB( KD, J1 ), INCA, D( J1 ),
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$ WORK( J1 ), KD1 )
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*
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* apply plane rotations from the left
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*
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IF( NR.GT.0 ) THEN
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IF( 2*KD-1.LT.NR ) THEN
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*
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* Dependent on the the number of diagonals either
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* SLARTV or SROT is used
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*
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DO 30 L = 1, KD - 1
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IF( J2+L.GT.N ) THEN
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NRT = NR - 1
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ELSE
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NRT = NR
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END IF
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IF( NRT.GT.0 )
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$ CALL SLARTV( NRT, AB( KD-L, J1+L ), INCA,
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$ AB( KD-L+1, J1+L ), INCA,
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$ D( J1 ), WORK( J1 ), KD1 )
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30 CONTINUE
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ELSE
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J1END = J1 + KD1*( NR-2 )
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IF( J1END.GE.J1 ) THEN
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DO 40 JIN = J1, J1END, KD1
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CALL SROT( KD-1, AB( KD-1, JIN+1 ), INCX,
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$ AB( KD, JIN+1 ), INCX,
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$ D( JIN ), WORK( JIN ) )
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40 CONTINUE
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END IF
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LEND = MIN( KDM1, N-J2 )
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LAST = J1END + KD1
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IF( LEND.GT.0 )
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$ CALL SROT( LEND, AB( KD-1, LAST+1 ), INCX,
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$ AB( KD, LAST+1 ), INCX, D( LAST ),
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$ WORK( LAST ) )
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END IF
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END IF
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*
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IF( WANTQ ) THEN
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*
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* accumulate product of plane rotations in Q
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*
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IF( INITQ ) THEN
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*
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* take advantage of the fact that Q was
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* initially the Identity matrix
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*
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IQEND = MAX( IQEND, J2 )
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I2 = MAX( 0, K-3 )
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IQAEND = 1 + I*KD
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IF( K.EQ.2 )
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$ IQAEND = IQAEND + KD
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IQAEND = MIN( IQAEND, IQEND )
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DO 50 J = J1, J2, KD1
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IBL = I - I2 / KDM1
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I2 = I2 + 1
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IQB = MAX( 1, J-IBL )
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NQ = 1 + IQAEND - IQB
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IQAEND = MIN( IQAEND+KD, IQEND )
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CALL SROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ),
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$ 1, D( J ), WORK( J ) )
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50 CONTINUE
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ELSE
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*
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DO 60 J = J1, J2, KD1
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CALL SROT( N, Q( 1, J-1 ), 1, Q( 1, J ), 1,
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$ D( J ), WORK( J ) )
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60 CONTINUE
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END IF
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*
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END IF
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*
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IF( J2+KDN.GT.N ) THEN
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*
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* adjust J2 to keep within the bounds of the matrix
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*
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NR = NR - 1
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J2 = J2 - KDN - 1
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END IF
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*
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DO 70 J = J1, J2, KD1
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*
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* create nonzero element a(j-1,j+kd) outside the band
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* and store it in WORK
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*
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WORK( J+KD ) = WORK( J )*AB( 1, J+KD )
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AB( 1, J+KD ) = D( J )*AB( 1, J+KD )
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70 CONTINUE
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80 CONTINUE
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90 CONTINUE
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END IF
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*
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IF( KD.GT.0 ) THEN
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*
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* copy off-diagonal elements to E
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*
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DO 100 I = 1, N - 1
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E( I ) = AB( KD, I+1 )
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100 CONTINUE
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ELSE
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*
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* set E to zero if original matrix was diagonal
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*
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DO 110 I = 1, N - 1
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E( I ) = ZERO
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110 CONTINUE
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END IF
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*
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* copy diagonal elements to D
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*
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DO 120 I = 1, N
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D( I ) = AB( KD1, I )
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120 CONTINUE
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*
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ELSE
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*
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IF( KD.GT.1 ) THEN
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*
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* Reduce to tridiagonal form, working with lower triangle
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*
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NR = 0
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J1 = KDN + 2
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J2 = 1
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*
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DO 210 I = 1, N - 2
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*
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* Reduce i-th column of matrix to tridiagonal form
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*
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DO 200 K = KDN + 1, 2, -1
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J1 = J1 + KDN
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J2 = J2 + KDN
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*
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IF( NR.GT.0 ) THEN
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*
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* generate plane rotations to annihilate nonzero
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* elements which have been created outside the band
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*
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CALL SLARGV( NR, AB( KD1, J1-KD1 ), INCA,
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$ WORK( J1 ), KD1, D( J1 ), KD1 )
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*
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* apply plane rotations from one side
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*
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*
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* Dependent on the the number of diagonals either
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* SLARTV or SROT is used
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*
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IF( NR.GT.2*KD-1 ) THEN
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DO 130 L = 1, KD - 1
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CALL SLARTV( NR, AB( KD1-L, J1-KD1+L ), INCA,
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$ AB( KD1-L+1, J1-KD1+L ), INCA,
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$ D( J1 ), WORK( J1 ), KD1 )
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130 CONTINUE
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ELSE
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JEND = J1 + KD1*( NR-1 )
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DO 140 JINC = J1, JEND, KD1
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CALL SROT( KDM1, AB( KD, JINC-KD ), INCX,
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$ AB( KD1, JINC-KD ), INCX,
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$ D( JINC ), WORK( JINC ) )
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140 CONTINUE
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END IF
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*
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END IF
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*
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IF( K.GT.2 ) THEN
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IF( K.LE.N-I+1 ) THEN
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*
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* generate plane rotation to annihilate a(i+k-1,i)
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* within the band
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*
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CALL SLARTG( AB( K-1, I ), AB( K, I ),
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$ D( I+K-1 ), WORK( I+K-1 ), TEMP )
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AB( K-1, I ) = TEMP
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*
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* apply rotation from the left
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*
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CALL SROT( K-3, AB( K-2, I+1 ), LDAB-1,
|
|
$ AB( K-1, I+1 ), LDAB-1, D( I+K-1 ),
|
|
$ WORK( I+K-1 ) )
|
|
END IF
|
|
NR = NR + 1
|
|
J1 = J1 - KDN - 1
|
|
END IF
|
|
*
|
|
* apply plane rotations from both sides to diagonal
|
|
* blocks
|
|
*
|
|
IF( NR.GT.0 )
|
|
$ CALL SLAR2V( NR, AB( 1, J1-1 ), AB( 1, J1 ),
|
|
$ AB( 2, J1-1 ), INCA, D( J1 ),
|
|
$ WORK( J1 ), KD1 )
|
|
*
|
|
* apply plane rotations from the right
|
|
*
|
|
*
|
|
* Dependent on the the number of diagonals either
|
|
* SLARTV or SROT is used
|
|
*
|
|
IF( NR.GT.0 ) THEN
|
|
IF( NR.GT.2*KD-1 ) THEN
|
|
DO 150 L = 1, KD - 1
|
|
IF( J2+L.GT.N ) THEN
|
|
NRT = NR - 1
|
|
ELSE
|
|
NRT = NR
|
|
END IF
|
|
IF( NRT.GT.0 )
|
|
$ CALL SLARTV( NRT, AB( L+2, J1-1 ), INCA,
|
|
$ AB( L+1, J1 ), INCA, D( J1 ),
|
|
$ WORK( J1 ), KD1 )
|
|
150 CONTINUE
|
|
ELSE
|
|
J1END = J1 + KD1*( NR-2 )
|
|
IF( J1END.GE.J1 ) THEN
|
|
DO 160 J1INC = J1, J1END, KD1
|
|
CALL SROT( KDM1, AB( 3, J1INC-1 ), 1,
|
|
$ AB( 2, J1INC ), 1, D( J1INC ),
|
|
$ WORK( J1INC ) )
|
|
160 CONTINUE
|
|
END IF
|
|
LEND = MIN( KDM1, N-J2 )
|
|
LAST = J1END + KD1
|
|
IF( LEND.GT.0 )
|
|
$ CALL SROT( LEND, AB( 3, LAST-1 ), 1,
|
|
$ AB( 2, LAST ), 1, D( LAST ),
|
|
$ WORK( LAST ) )
|
|
END IF
|
|
END IF
|
|
*
|
|
*
|
|
*
|
|
IF( WANTQ ) THEN
|
|
*
|
|
* accumulate product of plane rotations in Q
|
|
*
|
|
IF( INITQ ) THEN
|
|
*
|
|
* take advantage of the fact that Q was
|
|
* initially the Identity matrix
|
|
*
|
|
IQEND = MAX( IQEND, J2 )
|
|
I2 = MAX( 0, K-3 )
|
|
IQAEND = 1 + I*KD
|
|
IF( K.EQ.2 )
|
|
$ IQAEND = IQAEND + KD
|
|
IQAEND = MIN( IQAEND, IQEND )
|
|
DO 170 J = J1, J2, KD1
|
|
IBL = I - I2 / KDM1
|
|
I2 = I2 + 1
|
|
IQB = MAX( 1, J-IBL )
|
|
NQ = 1 + IQAEND - IQB
|
|
IQAEND = MIN( IQAEND+KD, IQEND )
|
|
CALL SROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ),
|
|
$ 1, D( J ), WORK( J ) )
|
|
170 CONTINUE
|
|
ELSE
|
|
*
|
|
DO 180 J = J1, J2, KD1
|
|
CALL SROT( N, Q( 1, J-1 ), 1, Q( 1, J ), 1,
|
|
$ D( J ), WORK( J ) )
|
|
180 CONTINUE
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( J2+KDN.GT.N ) THEN
|
|
*
|
|
* adjust J2 to keep within the bounds of the matrix
|
|
*
|
|
NR = NR - 1
|
|
J2 = J2 - KDN - 1
|
|
END IF
|
|
*
|
|
DO 190 J = J1, J2, KD1
|
|
*
|
|
* create nonzero element a(j+kd,j-1) outside the
|
|
* band and store it in WORK
|
|
*
|
|
WORK( J+KD ) = WORK( J )*AB( KD1, J )
|
|
AB( KD1, J ) = D( J )*AB( KD1, J )
|
|
190 CONTINUE
|
|
200 CONTINUE
|
|
210 CONTINUE
|
|
END IF
|
|
*
|
|
IF( KD.GT.0 ) THEN
|
|
*
|
|
* copy off-diagonal elements to E
|
|
*
|
|
DO 220 I = 1, N - 1
|
|
E( I ) = AB( 2, I )
|
|
220 CONTINUE
|
|
ELSE
|
|
*
|
|
* set E to zero if original matrix was diagonal
|
|
*
|
|
DO 230 I = 1, N - 1
|
|
E( I ) = ZERO
|
|
230 CONTINUE
|
|
END IF
|
|
*
|
|
* copy diagonal elements to D
|
|
*
|
|
DO 240 I = 1, N
|
|
D( I ) = AB( 1, I )
|
|
240 CONTINUE
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of SSBTRD
|
|
*
|
|
END
|