*> \brief SSPSVX 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 SSPSVX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, * LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER FACT, UPLO * INTEGER INFO, LDB, LDX, N, NRHS * REAL RCOND * .. * .. Array Arguments .. * INTEGER IPIV( * ), IWORK( * ) * REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ), * \$ FERR( * ), WORK( * ), X( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or *> A = L*D*L**T to compute the solution to a real system of linear *> equations A * X = B, where A is an N-by-N symmetric matrix stored *> in packed format 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 = 'N', the diagonal pivoting method is used to factor A as *> A = U * D * U**T, if UPLO = 'U', or *> A = L * D * L**T, if UPLO = 'L', *> where U (or L) is a product of permutation and unit upper (lower) *> triangular matrices and D is symmetric and block diagonal with *> 1-by-1 and 2-by-2 diagonal blocks. *> *> 2. If some D(i,i)=0, so that D is exactly singular, 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. *> *> 3. The system of equations is solved for X using the factored form *> of A. *> *> 4. Iterative refinement is applied to improve the computed solution *> matrix and calculate error bounds and backward error estimates *> for it. *> \endverbatim * * Arguments: * ========== * *> \param[in] FACT *> \verbatim *> FACT is CHARACTER*1 *> Specifies whether or not the factored form of A has been *> supplied on entry. *> = 'F': On entry, AFP and IPIV contain the factored form of *> A. AP, AFP and IPIV will not be modified. *> = 'N': The matrix A will be copied to AFP 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] 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] AP *> \verbatim *> AP is REAL array, dimension (N*(N+1)/2) *> The upper or lower triangle of the symmetric matrix A, packed *> columnwise in a linear array. The j-th column of A is stored *> in the array AP as follows: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. *> See below for further details. *> \endverbatim *> *> \param[in,out] AFP *> \verbatim *> AFP is REAL array, dimension (N*(N+1)/2) *> If FACT = 'F', then AFP is an input argument and on entry *> contains the block diagonal matrix D and the multipliers used *> to obtain the factor U or L from the factorization *> A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as *> a packed triangular matrix in the same storage format as A. *> *> If FACT = 'N', then AFP is an output argument and on exit *> contains the block diagonal matrix D and the multipliers used *> to obtain the factor U or L from the factorization *> A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as *> a packed triangular matrix in the same storage format as A. *> \endverbatim *> *> \param[in,out] IPIV *> \verbatim *> IPIV is INTEGER array, dimension (N) *> If FACT = 'F', then IPIV is an input argument and on entry *> contains details of the interchanges and the block structure *> of D, as determined by SSPTRF. *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were *> interchanged and D(k,k) is a 1-by-1 diagonal block. *> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and *> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) *> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = *> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were *> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. *> *> If FACT = 'N', then IPIV is an output argument and on exit *> contains details of the interchanges and the block structure *> of D, as determined by SSPTRF. *> \endverbatim *> *> \param[in] B *> \verbatim *> B is REAL array, dimension (LDB,NRHS) *> The N-by-NRHS right hand side matrix 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 REAL array, dimension (LDX,NRHS) *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix 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 REAL *> The estimate of the reciprocal condition number of the matrix *> A. 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 REAL 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 REAL 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 REAL 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: D(i,i) is exactly zero. The factorization *> has been completed but the factor D is exactly *> singular, so the solution and error bounds could *> not be computed. RCOND = 0 is returned. *> = N+1: D 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. * *> \date April 2012 * *> \ingroup realOTHERsolve * *> \par Further Details: * ===================== *> *> \verbatim *> *> The packed storage scheme is illustrated by the following example *> when N = 4, UPLO = 'U': *> *> Two-dimensional storage of the symmetric matrix A: *> *> a11 a12 a13 a14 *> a22 a23 a24 *> a33 a34 (aij = aji) *> a44 *> *> Packed storage of the upper triangle of A: *> *> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] *> \endverbatim *> * ===================================================================== SUBROUTINE SSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, \$ LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) * * -- LAPACK driver routine (version 3.7.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * April 2012 * * .. Scalar Arguments .. CHARACTER FACT, UPLO INTEGER INFO, LDB, LDX, N, NRHS REAL RCOND * .. * .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ), \$ FERR( * ), WORK( * ), X( LDX, * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO PARAMETER ( ZERO = 0.0E+0 ) * .. * .. Local Scalars .. LOGICAL NOFACT REAL ANORM * .. * .. External Functions .. LOGICAL LSAME REAL SLAMCH, SLANSP EXTERNAL LSAME, SLAMCH, SLANSP * .. * .. External Subroutines .. EXTERNAL SCOPY, SLACPY, SSPCON, SSPRFS, SSPTRF, SSPTRS, \$ XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 NOFACT = LSAME( FACT, 'N' ) IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN INFO = -1 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) \$ THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SSPSVX', -INFO ) RETURN END IF * IF( NOFACT ) THEN * * Compute the factorization A = U*D*U**T or A = L*D*L**T. * CALL SCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 ) CALL SSPTRF( UPLO, N, AFP, IPIV, 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 = SLANSP( 'I', UPLO, N, AP, WORK ) * * Compute the reciprocal of the condition number of A. * CALL SSPCON( UPLO, N, AFP, IPIV, ANORM, RCOND, WORK, IWORK, INFO ) * * Compute the solution vectors X. * CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL SSPTRS( UPLO, N, NRHS, AFP, IPIV, X, LDX, INFO ) * * Use iterative refinement to improve the computed solutions and * compute error bounds and backward error estimates for them. * CALL SSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, \$ BERR, WORK, IWORK, INFO ) * * Set INFO = N+1 if the matrix is singular to working precision. * IF( RCOND.LT.SLAMCH( 'Epsilon' ) ) \$ INFO = N + 1 * RETURN * * End of SSPSVX * END