--- rpl/lapack/lapack/zspsvx.f 2010/08/13 21:04:14 1.6
+++ rpl/lapack/lapack/zspsvx.f 2014/01/27 09:28:42 1.14
@@ -1,10 +1,286 @@
+*> \brief ZSPSVX 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 ZSPSVX + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
+* LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER FACT, UPLO
+* INTEGER INFO, LDB, LDX, N, NRHS
+* DOUBLE PRECISION RCOND
+* ..
+* .. Array Arguments ..
+* INTEGER IPIV( * )
+* DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
+* COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
+* $ X( LDX, * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
+*> A = L*D*L**T to compute the solution to a complex 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 COMPLEX*16 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 COMPLEX*16 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 ZSPTRF, 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 ZSPTRF, 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 ZSPTRF.
+*> 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 ZSPTRF.
+*> \endverbatim
+*>
+*> \param[in] B
+*> \verbatim
+*> B is COMPLEX*16 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 COMPLEX*16 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 DOUBLE PRECISION
+*> 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 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: 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 complex16OTHERsolve
+*
+*> \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 ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
$ LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
*
-* -- LAPACK driver routine (version 3.2) --
+* -- LAPACK driver routine (version 3.4.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2006
+* April 2012
*
* .. Scalar Arguments ..
CHARACTER FACT, UPLO
@@ -18,173 +294,6 @@
$ X( LDX, * )
* ..
*
-* Purpose
-* =======
-*
-* ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
-* A = L*D*L**T to compute the solution to a complex 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.
-*
-* Description
-* ===========
-*
-* 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.
-*
-* Arguments
-* =========
-*
-* FACT (input) 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.
-*
-* UPLO (input) CHARACTER*1
-* = 'U': Upper triangle of A is stored;
-* = 'L': Lower triangle of A is stored.
-*
-* N (input) INTEGER
-* The number of linear equations, i.e., the order of the
-* matrix A. N >= 0.
-*
-* NRHS (input) INTEGER
-* The number of right hand sides, i.e., the number of columns
-* of the matrices B and X. NRHS >= 0.
-*
-* AP (input) COMPLEX*16 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.
-*
-* AFP (input or output) COMPLEX*16 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 ZSPTRF, 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 ZSPTRF, stored as
-* a packed triangular matrix in the same storage format as A.
-*
-* IPIV (input or output) 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 ZSPTRF.
-* 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 ZSPTRF.
-*
-* B (input) COMPLEX*16 array, dimension (LDB,NRHS)
-* The N-by-NRHS right hand side matrix B.
-*
-* LDB (input) INTEGER
-* The leading dimension of the array B. LDB >= max(1,N).
-*
-* X (output) COMPLEX*16 array, dimension (LDX,NRHS)
-* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
-*
-* LDX (input) INTEGER
-* The leading dimension of the array X. LDX >= max(1,N).
-*
-* RCOND (output) DOUBLE PRECISION
-* 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.
-*
-* FERR (output) 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.
-*
-* BERR (output) 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).
-*
-* WORK (workspace) COMPLEX*16 array, dimension (2*N)
-*
-* RWORK (workspace) DOUBLE PRECISION array, dimension (N)
-*
-* INFO (output) 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.
-*
-* Further Details
-* ===============
-*
-* 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 ]
-*
* =====================================================================
*
* .. Parameters ..
@@ -234,7 +343,7 @@
*
IF( NOFACT ) THEN
*
-* Compute the factorization A = U*D*U' or A = L*D*L'.
+* Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
CALL ZSPTRF( UPLO, N, AFP, IPIV, INFO )