*> \brief ZPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZPBSVX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, * EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, * WORK, RWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER EQUED, FACT, UPLO * INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS * DOUBLE PRECISION RCOND * .. * .. Array Arguments .. * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * ) * COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), * $ WORK( * ), X( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to *> compute the solution to a complex system of linear equations *> A * X = B, *> where A is an N-by-N Hermitian 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**H * U, if UPLO = 'U', or *> A = L * L**H, 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 COMPLEX*16 array, dimension (LDAB,N) *> On entry, the upper or lower triangle of the Hermitian 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 COMPLEX*16 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**H *U or A = L*L**H 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**H *U or A = L*L**H. *> *> 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**H *U or A = L*L**H 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension (2*N) *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION 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 complex16OTHERsolve * *> \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 Hermitian 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 ZPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, $ WORK, RWORK, 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 .. DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * ) COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), $ 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, ZLANHB EXTERNAL LSAME, DLAMCH, ZLANHB * .. * .. External Subroutines .. EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAQHB, ZPBCON, ZPBEQU, $ ZPBRFS, ZPBTRF, ZPBTRS * .. * .. 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( 'ZPBSVX', -INFO ) RETURN END IF * IF( EQUIL ) THEN * * Compute row and column scalings to equilibrate the matrix A. * CALL ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN * * Equilibrate the matrix. * CALL ZLAQHB( 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**H *U or A = L*L**H. * IF( UPPER ) THEN DO 40 J = 1, N J1 = MAX( J-KD, 1 ) CALL ZCOPY( 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 ZCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 ) 50 CONTINUE END IF * CALL ZPBTRF( 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 = ZLANHB( '1', UPLO, N, KD, AB, LDAB, RWORK ) * * Compute the reciprocal of the condition number of A. * CALL ZPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, RWORK, $ INFO ) * * Compute the solution matrix X. * CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL ZPBTRS( 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 ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, $ LDX, FERR, BERR, WORK, RWORK, 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 ZPBSVX * END