phynexis-deps/lapack
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
Files
f86d08bae6c46685e0f759850e82dc0a8fa00a21
lapack/SRC/dpbsvx.f
T
Julie e18d437924 Making LAPACK's code eternal... no more version and date in source files. 
GitHub is now enabling us to track accurately version and date.
No need for this anymore.
2021-03-25 10:16:58 -07:00

543 lines
18 KiB
FortranFixed
Raw Blame History

*> \brief <b> DPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
* EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, FACT, UPLO
* INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
* $ BERR( * ), FERR( * ), S( * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
*> compute the solution to a real system of linear equations
*> A * X = B,
*> where A is an N-by-N symmetric positive definite band matrix and X
*> and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*> the system:
*> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
*> Whether or not the system will be equilibrated depends on the
*> scaling of the matrix A, but if equilibration is used, A is
*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*>
*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
*> factor the matrix A (after equilibration if FACT = 'E') as
*> A = U**T * U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular band matrix, and L is a lower
*> triangular band matrix.
*>
*> 3. If the leading i-by-i principal minor is not positive definite,
*> then the routine returns with INFO = i. Otherwise, the factored
*> form of A is used to estimate the condition number of the matrix
*> A. If the reciprocal of the condition number is less than machine
*> precision, INFO = N+1 is returned as a warning, but the routine
*> still goes on to solve for X and compute error bounds as
*> described below.
*>
*> 4. The system of equations is solved for X using the factored form
*> of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*> diag(S) so that it solves the original system before
*> equilibration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of the matrix A is
*> supplied on entry, and if not, whether the matrix A should be
*> equilibrated before it is factored.
*> = 'F': On entry, AFB contains the factored form of A.
*> If EQUED = 'Y', the matrix A has been equilibrated
*> with scaling factors given by S. AB and AFB will not
*> be modified.
*> = 'N': The matrix A will be copied to AFB and factored.
*> = 'E': The matrix A will be equilibrated if necessary, then
*> copied to AFB and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right-hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION 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, except
*> if FACT = 'F' and EQUED = 'Y', then A must contain the
*> equilibrated matrix diag(S)*A*diag(S). 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).
*> See below for further details.
*>
*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
*> diag(S)*A*diag(S).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array A. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] AFB
*> \verbatim
*> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
*> If FACT = 'F', then AFB is an input argument and on entry
*> contains 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 (see AB). If EQUED = 'Y',
*> then AFB is the factored form of the equilibrated matrix A.
*>
*> If FACT = 'N', then AFB is an output argument and on exit
*> returns the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T.
*>
*> If FACT = 'E', then AFB is an output argument and on exit
*> returns the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T of the equilibrated
*> matrix A (see the description of A for the form of the
*> equilibrated matrix).
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*> LDAFB is INTEGER
*> The leading dimension of the array AFB. LDAFB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies the form of equilibration that was done.
*> = 'N': No equilibration (always true if FACT = 'N').
*> = 'Y': Equilibration was done, i.e., A has been replaced by
*> diag(S) * A * diag(S).
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
*> output argument.
*> \endverbatim
*>
*> \param[in,out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The scale factors for A; not accessed if EQUED = 'N'. S is
*> an input argument if FACT = 'F'; otherwise, S is an output
*> argument. If FACT = 'F' and EQUED = 'Y', each element of S
*> must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
*> B is overwritten by diag(S) * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
*> the original system of equations. Note that if EQUED = 'Y',
*> A and B are modified on exit, and the solution to the
*> equilibrated system is inv(diag(S))*X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The estimate of the reciprocal condition number of the matrix
*> A after equilibration (if done). If RCOND is less than the
*> machine precision (in particular, if RCOND = 0), the matrix
*> is singular to working precision. This condition is
*> indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: the leading minor of order i of A is
*> not positive definite, so the factorization
*> could not be completed, and the solution has not
*> been computed. RCOND = 0 is returned.
*> = N+1: U is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERsolve
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> N = 6, KD = 2, and UPLO = 'U':
*>
*> Two-dimensional storage of the symmetric matrix A:
*>
*> a11 a12 a13
*> a22 a23 a24
*> a33 a34 a35
*> a44 a45 a46
*> a55 a56
*> (aij=conjg(aji)) a66
*>
*> Band storage of the upper triangle of A:
*>
*> * * a13 a24 a35 a46
*> * a12 a23 a34 a45 a56
*> a11 a22 a33 a44 a55 a66
*>
*> Similarly, if UPLO = 'L' the format of A is as follows:
*>
*> a11 a22 a33 a44 a55 a66
*> a21 a32 a43 a54 a65 *
*> a31 a42 a53 a64 * *
*>
*> Array elements marked * are not used by the routine.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
$ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
$ WORK, IWORK, INFO )
*
* -- LAPACK driver 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 EQUED, FACT, UPLO
INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), S( * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL EQUIL, NOFACT, RCEQU, UPPER
INTEGER I, INFEQU, J, J1, J2
DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSB
EXTERNAL LSAME, DLAMCH, DLANSB
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DLAQSB, DPBCON, DPBEQU, DPBRFS,
$ DPBTRF, DPBTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
UPPER = LSAME( UPLO, 'U' )
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
RCEQU = .FALSE.
ELSE
RCEQU = LSAME( EQUED, 'Y' )
SMLNUM = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
END IF
*
* Test the input parameters.
*
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
$ THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KD.LT.0 ) THEN
INFO = -4
ELSE IF( NRHS.LT.0 ) THEN
INFO = -5
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -7
ELSE IF( LDAFB.LT.KD+1 ) THEN
INFO = -9
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
$ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
INFO = -10
ELSE
IF( RCEQU ) THEN
SMIN = BIGNUM
SMAX = ZERO
DO 10 J = 1, N
SMIN = MIN( SMIN, S( J ) )
SMAX = MAX( SMAX, S( J ) )
10 CONTINUE
IF( SMIN.LE.ZERO ) THEN
INFO = -11
ELSE IF( N.GT.0 ) THEN
SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
ELSE
SCOND = ONE
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -13
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -15
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBSVX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
RCEQU = LSAME( EQUED, 'Y' )
END IF
END IF
*
* Scale the right-hand side.
*
IF( RCEQU ) THEN
DO 30 J = 1, NRHS
DO 20 I = 1, N
B( I, J ) = S( I )*B( I, J )
20 CONTINUE
30 CONTINUE
END IF
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the Cholesky factorization A = U**T *U or A = L*L**T.
*
IF( UPPER ) THEN
DO 40 J = 1, N
J1 = MAX( J-KD, 1 )
CALL DCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
$ AFB( KD+1-J+J1, J ), 1 )
40 CONTINUE
ELSE
DO 50 J = 1, N
J2 = MIN( J+KD, N )
CALL DCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
50 CONTINUE
END IF
*
CALL DPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 )THEN
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A.
*
ANORM = DLANSB( '1', UPLO, N, KD, AB, LDAB, WORK )
*
* Compute the reciprocal of the condition number of A.
*
CALL DPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, IWORK,
$ INFO )
*
* Compute the solution matrix X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solution and
* compute error bounds and backward error estimates for it.
*
CALL DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
$ LDX, FERR, BERR, WORK, IWORK, INFO )
*
* Transform the solution matrix X to a solution of the original
* system.
*
IF( RCEQU ) THEN
DO 70 J = 1, NRHS
DO 60 I = 1, N
X( I, J ) = S( I )*X( I, J )
60 CONTINUE
70 CONTINUE
DO 80 J = 1, NRHS
FERR( J ) = FERR( J ) / SCOND
80 CONTINUE
END IF
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of DPBSVX
*
END
Reference in New Issue View Git Blame Copy Permalink