version 1.6, 2010/08/13 21:03:43
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version 1.18, 2023/08/07 08:38:47
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*> \brief <b> DGBSVX computes the solution to system of linear equations A * X = B for GB matrices</b> |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download DGBSVX + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbsvx.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbsvx.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbsvx.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, |
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* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, |
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* RCOND, FERR, BERR, WORK, IWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER EQUED, FACT, TRANS |
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* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS |
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* DOUBLE PRECISION RCOND |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IPIV( * ), IWORK( * ) |
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* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), |
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* $ BERR( * ), C( * ), FERR( * ), R( * ), |
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* $ WORK( * ), X( LDX, * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> DGBSVX uses the LU factorization to compute the solution to a real |
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*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B, |
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*> where A is a band matrix of order N with KL subdiagonals and KU |
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*> superdiagonals, and X and B are N-by-NRHS matrices. |
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*> |
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*> Error bounds on the solution and a condition estimate are also |
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*> provided. |
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*> \endverbatim |
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* |
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*> \par Description: |
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* ================= |
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*> |
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*> \verbatim |
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*> |
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*> The following steps are performed by this subroutine: |
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*> |
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*> 1. If FACT = 'E', real scaling factors are computed to equilibrate |
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*> the system: |
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*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B |
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*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B |
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*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B |
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*> Whether or not the system will be equilibrated depends on the |
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*> scaling of the matrix A, but if equilibration is used, A is |
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*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') |
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*> or diag(C)*B (if TRANS = 'T' or 'C'). |
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*> |
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*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the |
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*> matrix A (after equilibration if FACT = 'E') as |
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*> A = L * U, |
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*> where L is a product of permutation and unit lower triangular |
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*> matrices with KL subdiagonals, and U is upper triangular with |
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*> KL+KU superdiagonals. |
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*> |
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*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine |
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*> returns with INFO = i. Otherwise, the factored form of A is used |
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*> to estimate the condition number of the matrix A. If the |
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*> reciprocal of the condition number is less than machine precision, |
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*> INFO = N+1 is returned as a warning, but the routine still goes on |
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*> to solve for X and compute error bounds as described below. |
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*> |
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*> 4. The system of equations is solved for X using the factored form |
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*> of A. |
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*> |
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*> 5. Iterative refinement is applied to improve the computed solution |
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*> matrix and calculate error bounds and backward error estimates |
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*> for it. |
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*> |
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*> 6. If equilibration was used, the matrix X is premultiplied by |
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*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so |
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*> that it solves the original system before equilibration. |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] FACT |
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*> \verbatim |
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*> FACT is CHARACTER*1 |
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*> Specifies whether or not the factored form of the matrix A is |
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*> supplied on entry, and if not, whether the matrix A should be |
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*> equilibrated before it is factored. |
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*> = 'F': On entry, AFB and IPIV contain the factored form of |
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*> A. If EQUED is not 'N', the matrix A has been |
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*> equilibrated with scaling factors given by R and C. |
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*> AB, AFB, and IPIV are not modified. |
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*> = 'N': The matrix A will be copied to AFB and factored. |
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*> = 'E': The matrix A will be equilibrated if necessary, then |
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*> copied to AFB and factored. |
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*> \endverbatim |
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*> |
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*> \param[in] TRANS |
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*> \verbatim |
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*> TRANS is CHARACTER*1 |
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*> Specifies the form of the system of equations. |
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*> = 'N': A * X = B (No transpose) |
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*> = 'T': A**T * X = B (Transpose) |
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*> = 'C': A**H * X = B (Transpose) |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The number of linear equations, i.e., the order of the |
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*> matrix A. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] KL |
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*> \verbatim |
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*> KL is INTEGER |
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*> The number of subdiagonals within the band of A. KL >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] KU |
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*> \verbatim |
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*> KU is INTEGER |
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*> The number of superdiagonals within the band of A. KU >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] NRHS |
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*> \verbatim |
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*> NRHS is INTEGER |
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*> The number of right hand sides, i.e., the number of columns |
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*> of the matrices B and X. NRHS >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in,out] AB |
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*> \verbatim |
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*> AB is DOUBLE PRECISION array, dimension (LDAB,N) |
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*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1. |
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*> The j-th column of A is stored in the j-th column of the |
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*> array AB as follows: |
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*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) |
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*> |
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*> If FACT = 'F' and EQUED is not 'N', then A must have been |
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*> equilibrated by the scaling factors in R and/or C. AB is not |
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*> modified if FACT = 'F' or 'N', or if FACT = 'E' and |
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*> EQUED = 'N' on exit. |
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*> |
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*> On exit, if EQUED .ne. 'N', A is scaled as follows: |
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*> EQUED = 'R': A := diag(R) * A |
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*> EQUED = 'C': A := A * diag(C) |
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*> EQUED = 'B': A := diag(R) * A * diag(C). |
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*> \endverbatim |
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*> |
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*> \param[in] LDAB |
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*> \verbatim |
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*> LDAB is INTEGER |
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*> The leading dimension of the array AB. LDAB >= KL+KU+1. |
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*> \endverbatim |
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*> |
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*> \param[in,out] AFB |
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*> \verbatim |
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*> AFB is DOUBLE PRECISION array, dimension (LDAFB,N) |
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*> If FACT = 'F', then AFB is an input argument and on entry |
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*> contains details of the LU factorization of the band matrix |
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*> A, as computed by DGBTRF. U is stored as an upper triangular |
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*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, |
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*> and the multipliers used during the factorization are stored |
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*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is |
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*> the factored form of the equilibrated matrix A. |
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*> |
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*> If FACT = 'N', then AFB is an output argument and on exit |
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*> returns details of the LU factorization of A. |
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*> |
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*> If FACT = 'E', then AFB is an output argument and on exit |
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*> returns details of the LU factorization of the equilibrated |
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*> matrix A (see the description of AB for the form of the |
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*> equilibrated matrix). |
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*> \endverbatim |
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*> |
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*> \param[in] LDAFB |
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*> \verbatim |
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*> LDAFB is INTEGER |
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*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. |
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*> \endverbatim |
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*> |
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*> \param[in,out] IPIV |
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*> \verbatim |
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*> IPIV is INTEGER array, dimension (N) |
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*> If FACT = 'F', then IPIV is an input argument and on entry |
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*> contains the pivot indices from the factorization A = L*U |
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*> as computed by DGBTRF; row i of the matrix was interchanged |
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*> with row IPIV(i). |
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*> |
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*> If FACT = 'N', then IPIV is an output argument and on exit |
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*> contains the pivot indices from the factorization A = L*U |
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*> of the original matrix A. |
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*> |
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*> If FACT = 'E', then IPIV is an output argument and on exit |
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*> contains the pivot indices from the factorization A = L*U |
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*> of the equilibrated matrix A. |
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*> \endverbatim |
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*> |
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*> \param[in,out] EQUED |
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*> \verbatim |
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*> EQUED is CHARACTER*1 |
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*> Specifies the form of equilibration that was done. |
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*> = 'N': No equilibration (always true if FACT = 'N'). |
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*> = 'R': Row equilibration, i.e., A has been premultiplied by |
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*> diag(R). |
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*> = 'C': Column equilibration, i.e., A has been postmultiplied |
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*> by diag(C). |
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*> = 'B': Both row and column equilibration, i.e., A has been |
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*> replaced by diag(R) * A * diag(C). |
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*> EQUED is an input argument if FACT = 'F'; otherwise, it is an |
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*> output argument. |
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*> \endverbatim |
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*> |
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*> \param[in,out] R |
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*> \verbatim |
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*> R is DOUBLE PRECISION array, dimension (N) |
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*> The row scale factors for A. If EQUED = 'R' or 'B', A is |
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*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R |
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*> is not accessed. R is an input argument if FACT = 'F'; |
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*> otherwise, R is an output argument. If FACT = 'F' and |
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*> EQUED = 'R' or 'B', each element of R must be positive. |
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*> \endverbatim |
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*> |
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*> \param[in,out] C |
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*> \verbatim |
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*> C is DOUBLE PRECISION array, dimension (N) |
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*> The column scale factors for A. If EQUED = 'C' or 'B', A is |
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*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C |
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*> is not accessed. C is an input argument if FACT = 'F'; |
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*> otherwise, C is an output argument. If FACT = 'F' and |
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*> EQUED = 'C' or 'B', each element of C must be positive. |
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*> \endverbatim |
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*> |
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*> \param[in,out] B |
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*> \verbatim |
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*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) |
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*> On entry, the right hand side matrix B. |
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*> On exit, |
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*> if EQUED = 'N', B is not modified; |
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*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by |
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*> diag(R)*B; |
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*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is |
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*> overwritten by diag(C)*B. |
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*> \endverbatim |
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*> |
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*> \param[in] LDB |
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*> \verbatim |
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*> LDB is INTEGER |
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*> The leading dimension of the array B. LDB >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] X |
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*> \verbatim |
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*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) |
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*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X |
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*> to the original system of equations. Note that A and B are |
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*> modified on exit if EQUED .ne. 'N', and the solution to the |
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*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and |
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*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' |
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*> and EQUED = 'R' or 'B'. |
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*> \endverbatim |
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*> |
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*> \param[in] LDX |
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*> \verbatim |
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*> LDX is INTEGER |
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*> The leading dimension of the array X. LDX >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] RCOND |
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*> \verbatim |
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*> RCOND is DOUBLE PRECISION |
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*> The estimate of the reciprocal condition number of the matrix |
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*> A after equilibration (if done). If RCOND is less than the |
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*> machine precision (in particular, if RCOND = 0), the matrix |
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*> is singular to working precision. This condition is |
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*> indicated by a return code of INFO > 0. |
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*> \endverbatim |
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*> |
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*> \param[out] FERR |
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*> \verbatim |
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*> FERR is DOUBLE PRECISION array, dimension (NRHS) |
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*> The estimated forward error bound for each solution vector |
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*> X(j) (the j-th column of the solution matrix X). |
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*> If XTRUE is the true solution corresponding to X(j), FERR(j) |
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*> is an estimated upper bound for the magnitude of the largest |
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*> element in (X(j) - XTRUE) divided by the magnitude of the |
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*> largest element in X(j). The estimate is as reliable as |
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*> the estimate for RCOND, and is almost always a slight |
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*> overestimate of the true error. |
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*> \endverbatim |
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*> |
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*> \param[out] BERR |
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*> \verbatim |
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*> BERR is DOUBLE PRECISION array, dimension (NRHS) |
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*> The componentwise relative backward error of each solution |
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*> vector X(j) (i.e., the smallest relative change in |
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*> any element of A or B that makes X(j) an exact solution). |
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*> \endverbatim |
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*> |
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*> \param[out] WORK |
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*> \verbatim |
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*> WORK is DOUBLE PRECISION array, dimension (3*N) |
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*> On exit, WORK(1) contains the reciprocal pivot growth |
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*> factor norm(A)/norm(U). The "max absolute element" norm is |
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*> used. If WORK(1) is much less than 1, then the stability |
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*> of the LU factorization of the (equilibrated) matrix A |
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*> could be poor. This also means that the solution X, condition |
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*> estimator RCOND, and forward error bound FERR could be |
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*> unreliable. If factorization fails with 0<INFO<=N, then |
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*> WORK(1) contains the reciprocal pivot growth factor for the |
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*> leading INFO columns of A. |
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*> \endverbatim |
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*> |
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*> \param[out] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: successful exit |
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*> < 0: if INFO = -i, the i-th argument had an illegal value |
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*> > 0: if INFO = i, and i is |
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*> <= N: U(i,i) is exactly zero. The factorization |
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*> has been completed, but the factor U is exactly |
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*> singular, so the solution and error bounds |
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*> could not be computed. RCOND = 0 is returned. |
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*> = N+1: U is nonsingular, but RCOND is less than machine |
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*> precision, meaning that the matrix is singular |
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*> to working precision. Nevertheless, the |
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*> solution and error bounds are computed because |
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*> there are a number of situations where the |
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*> computed solution can be more accurate than the |
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*> value of RCOND would suggest. |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \ingroup doubleGBsolve |
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* |
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* ===================================================================== |
SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, |
SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, |
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, |
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, |
$ RCOND, FERR, BERR, WORK, IWORK, INFO ) |
$ RCOND, FERR, BERR, WORK, IWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver routine -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER EQUED, FACT, TRANS |
CHARACTER EQUED, FACT, TRANS |
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$ WORK( * ), X( LDX, * ) |
$ WORK( * ), X( LDX, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DGBSVX uses the LU factorization to compute the solution to a real |
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* system of linear equations A * X = B, A**T * X = B, or A**H * X = B, |
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* where A is a band matrix of order N with KL subdiagonals and KU |
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* superdiagonals, and X and B are N-by-NRHS matrices. |
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* |
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* Error bounds on the solution and a condition estimate are also |
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* provided. |
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* |
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* Description |
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* =========== |
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* |
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* The following steps are performed by this subroutine: |
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* |
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* 1. If FACT = 'E', real scaling factors are computed to equilibrate |
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* the system: |
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* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B |
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* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B |
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* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B |
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* Whether or not the system will be equilibrated depends on the |
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* scaling of the matrix A, but if equilibration is used, A is |
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* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') |
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* or diag(C)*B (if TRANS = 'T' or 'C'). |
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* |
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* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the |
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* matrix A (after equilibration if FACT = 'E') as |
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* A = L * U, |
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* where L is a product of permutation and unit lower triangular |
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* matrices with KL subdiagonals, and U is upper triangular with |
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* KL+KU superdiagonals. |
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* |
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* 3. If some U(i,i)=0, so that U is exactly singular, then the routine |
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* returns with INFO = i. Otherwise, the factored form of A is used |
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* to estimate the condition number of the matrix A. If the |
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* reciprocal of the condition number is less than machine precision, |
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* INFO = N+1 is returned as a warning, but the routine still goes on |
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* to solve for X and compute error bounds as described below. |
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* |
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* 4. The system of equations is solved for X using the factored form |
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* of A. |
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* |
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* 5. Iterative refinement is applied to improve the computed solution |
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* matrix and calculate error bounds and backward error estimates |
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* for it. |
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* |
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* 6. If equilibration was used, the matrix X is premultiplied by |
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* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so |
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* that it solves the original system before equilibration. |
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* |
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* Arguments |
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* ========= |
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* |
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* FACT (input) CHARACTER*1 |
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* Specifies whether or not the factored form of the matrix A is |
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* supplied on entry, and if not, whether the matrix A should be |
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* equilibrated before it is factored. |
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* = 'F': On entry, AFB and IPIV contain the factored form of |
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* A. If EQUED is not 'N', the matrix A has been |
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* equilibrated with scaling factors given by R and C. |
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* AB, AFB, and IPIV are not modified. |
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* = 'N': The matrix A will be copied to AFB and factored. |
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* = 'E': The matrix A will be equilibrated if necessary, then |
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* copied to AFB and factored. |
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* |
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* TRANS (input) CHARACTER*1 |
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* Specifies the form of the system of equations. |
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* = 'N': A * X = B (No transpose) |
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* = 'T': A**T * X = B (Transpose) |
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* = 'C': A**H * X = B (Transpose) |
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* |
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* N (input) INTEGER |
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* The number of linear equations, i.e., the order of the |
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* matrix A. N >= 0. |
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* |
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* KL (input) INTEGER |
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* The number of subdiagonals within the band of A. KL >= 0. |
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* |
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* KU (input) INTEGER |
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* The number of superdiagonals within the band of A. KU >= 0. |
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* |
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* NRHS (input) INTEGER |
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* The number of right hand sides, i.e., the number of columns |
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* of the matrices B and X. NRHS >= 0. |
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* |
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* AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) |
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* On entry, the matrix A in band storage, in rows 1 to KL+KU+1. |
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* The j-th column of A is stored in the j-th column of the |
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* array AB as follows: |
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* AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) |
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* |
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* If FACT = 'F' and EQUED is not 'N', then A must have been |
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* equilibrated by the scaling factors in R and/or C. AB is not |
|
* modified if FACT = 'F' or 'N', or if FACT = 'E' and |
|
* EQUED = 'N' on exit. |
|
* |
|
* On exit, if EQUED .ne. 'N', A is scaled as follows: |
|
* EQUED = 'R': A := diag(R) * A |
|
* EQUED = 'C': A := A * diag(C) |
|
* EQUED = 'B': A := diag(R) * A * diag(C). |
|
* |
|
* LDAB (input) INTEGER |
|
* The leading dimension of the array AB. LDAB >= KL+KU+1. |
|
* |
|
* AFB (input or output) DOUBLE PRECISION array, dimension (LDAFB,N) |
|
* If FACT = 'F', then AFB is an input argument and on entry |
|
* contains details of the LU factorization of the band matrix |
|
* A, as computed by DGBTRF. U is stored as an upper triangular |
|
* band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, |
|
* and the multipliers used during the factorization are stored |
|
* in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is |
|
* the factored form of the equilibrated matrix A. |
|
* |
|
* If FACT = 'N', then AFB is an output argument and on exit |
|
* returns details of the LU factorization of A. |
|
* |
|
* If FACT = 'E', then AFB is an output argument and on exit |
|
* returns details of the LU factorization of the equilibrated |
|
* matrix A (see the description of AB for the form of the |
|
* equilibrated matrix). |
|
* |
|
* LDAFB (input) INTEGER |
|
* The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. |
|
* |
|
* IPIV (input or output) INTEGER array, dimension (N) |
|
* If FACT = 'F', then IPIV is an input argument and on entry |
|
* contains the pivot indices from the factorization A = L*U |
|
* as computed by DGBTRF; row i of the matrix was interchanged |
|
* with row IPIV(i). |
|
* |
|
* If FACT = 'N', then IPIV is an output argument and on exit |
|
* contains the pivot indices from the factorization A = L*U |
|
* of the original matrix A. |
|
* |
|
* If FACT = 'E', then IPIV is an output argument and on exit |
|
* contains the pivot indices from the factorization A = L*U |
|
* of the equilibrated matrix A. |
|
* |
|
* EQUED (input or output) CHARACTER*1 |
|
* Specifies the form of equilibration that was done. |
|
* = 'N': No equilibration (always true if FACT = 'N'). |
|
* = 'R': Row equilibration, i.e., A has been premultiplied by |
|
* diag(R). |
|
* = 'C': Column equilibration, i.e., A has been postmultiplied |
|
* by diag(C). |
|
* = 'B': Both row and column equilibration, i.e., A has been |
|
* replaced by diag(R) * A * diag(C). |
|
* EQUED is an input argument if FACT = 'F'; otherwise, it is an |
|
* output argument. |
|
* |
|
* R (input or output) DOUBLE PRECISION array, dimension (N) |
|
* The row scale factors for A. If EQUED = 'R' or 'B', A is |
|
* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R |
|
* is not accessed. R is an input argument if FACT = 'F'; |
|
* otherwise, R is an output argument. If FACT = 'F' and |
|
* EQUED = 'R' or 'B', each element of R must be positive. |
|
* |
|
* C (input or output) DOUBLE PRECISION array, dimension (N) |
|
* The column scale factors for A. If EQUED = 'C' or 'B', A is |
|
* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C |
|
* is not accessed. C is an input argument if FACT = 'F'; |
|
* otherwise, C is an output argument. If FACT = 'F' and |
|
* EQUED = 'C' or 'B', each element of C must be positive. |
|
* |
|
* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) |
|
* On entry, the right hand side matrix B. |
|
* On exit, |
|
* if EQUED = 'N', B is not modified; |
|
* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by |
|
* diag(R)*B; |
|
* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is |
|
* overwritten by diag(C)*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 A and B are |
|
* modified on exit if EQUED .ne. 'N', and the solution to the |
|
* equilibrated system is inv(diag(C))*X if TRANS = 'N' and |
|
* EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' |
|
* and EQUED = 'R' or 'B'. |
|
* |
|
* 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/output) DOUBLE PRECISION array, dimension (3*N) |
|
* On exit, WORK(1) contains the reciprocal pivot growth |
|
* factor norm(A)/norm(U). The "max absolute element" norm is |
|
* used. If WORK(1) is much less than 1, then the stability |
|
* of the LU factorization of the (equilibrated) matrix A |
|
* could be poor. This also means that the solution X, condition |
|
* estimator RCOND, and forward error bound FERR could be |
|
* unreliable. If factorization fails with 0<INFO<=N, then |
|
* WORK(1) contains the reciprocal pivot growth factor for the |
|
* leading INFO columns of A. |
|
* |
|
* 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: U(i,i) is exactly zero. The factorization |
|
* has been completed, but the factor U is exactly |
|
* singular, so the solution and error bounds |
|
* could not be 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. |
|
* |
|
* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |