--- rpl/lapack/lapack/dpbsvx.f 2011/07/22 07:38:09 1.8
+++ rpl/lapack/lapack/dpbsvx.f 2011/11/21 20:43:01 1.9
@@ -1,11 +1,352 @@
+*> \brief DPBSVX 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 DPBSVX + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \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 or output) 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 or output) 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 or output) 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.
+*
+*> \date November 2011
+*
+*> \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 (version 3.3.1) --
+* -- LAPACK driver routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* -- April 2011 --
+* November 2011
*
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, UPLO
@@ -19,225 +360,6 @@
$ X( LDX, * )
* ..
*
-* Purpose
-* =======
-*
-* 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.
-*
-* Description
-* ===========
-*
-* 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.
-*
-* Arguments
-* =========
-*
-* FACT (input) 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.
-*
-* 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.
-*
-* KD (input) INTEGER
-* The number of superdiagonals of the matrix A if UPLO = 'U',
-* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
-*
-* NRHS (input) INTEGER
-* The number of right-hand sides, i.e., the number of columns
-* of the matrices B and X. NRHS >= 0.
-*
-* AB (input/output) 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).
-*
-* LDAB (input) INTEGER
-* The leading dimension of the array A. LDAB >= KD+1.
-*
-* AFB (input or output) 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).
-*
-* LDAFB (input) INTEGER
-* The leading dimension of the array AFB. LDAFB >= KD+1.
-*
-* EQUED (input or output) 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.
-*
-* S (input or output) 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.
-*
-* B (input/output) 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.
-*
-* LDB (input) INTEGER
-* The leading dimension of the array B. LDB >= max(1,N).
-*
-* X (output) 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.
-*
-* 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 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.
-*
-* 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) DOUBLE PRECISION array, dimension (3*N)
-*
-* IWORK (workspace) INTEGER 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: 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.
-*
-* Further Details
-* ===============
-*
-* 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.
-*
* =====================================================================
*
* .. Parameters ..