642 lines
22 KiB
C
642 lines
22 KiB
C
#include "blaswrap.h"
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#ifdef _cpluscplus
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extern "C" {
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#endif
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#include "f2c.h"
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/* Subroutine */ int dgbsvx_(char *fact, char *trans, integer *n, integer *kl,
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integer *ku, integer *nrhs, doublereal *ab, integer *ldab,
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doublereal *afb, integer *ldafb, integer *ipiv, char *equed,
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doublereal *r__, doublereal *c__, doublereal *b, integer *ldb,
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doublereal *x, integer *ldx, doublereal *rcond, doublereal *ferr,
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doublereal *berr, doublereal *work, integer *iwork, integer *info)
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{
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/* -- LAPACK driver routine (version 3.0) --
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Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
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Courant Institute, Argonne National Lab, and Rice University
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June 30, 1999
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Purpose
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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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Error bounds on the solution and a condition estimate are also
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provided.
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Description
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===========
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The following steps are performed by this subroutine:
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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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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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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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4. The system of equations is solved for X using the factored form
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of A.
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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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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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Arguments
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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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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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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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KL (input) INTEGER
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The number of subdiagonals within the band of A. KL >= 0.
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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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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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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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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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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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LDAB (input) INTEGER
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The leading dimension of the array AB. LDAB >= KL+KU+1.
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AFB (input or output) 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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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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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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LDAFB (input) INTEGER
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The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
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IPIV (input or output) 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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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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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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EQUED (input or output) 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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R (input or output) 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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C (input or output) 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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B (input/output) 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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LDB (input) INTEGER
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The leading dimension of the array B. LDB >= max(1,N).
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X (output) 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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LDX (input) INTEGER
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The leading dimension of the array X. LDX >= max(1,N).
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RCOND (output) 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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FERR (output) 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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BERR (output) 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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WORK (workspace/output) 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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IWORK (workspace) INTEGER array, dimension (N)
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INFO (output) 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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=====================================================================
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Parameter adjustments */
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/* Table of constant values */
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static integer c__1 = 1;
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/* System generated locals */
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integer ab_dim1, ab_offset, afb_dim1, afb_offset, b_dim1, b_offset,
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x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5;
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doublereal d__1, d__2, d__3;
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/* Local variables */
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static doublereal amax;
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static char norm[1];
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static integer i__, j;
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extern logical lsame_(char *, char *);
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static doublereal rcmin, rcmax, anorm;
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extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
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doublereal *, integer *);
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static logical equil;
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static integer j1, j2;
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extern doublereal dlamch_(char *), dlangb_(char *, integer *,
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integer *, integer *, doublereal *, integer *, doublereal *);
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extern /* Subroutine */ int dlaqgb_(integer *, integer *, integer *,
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integer *, doublereal *, integer *, doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, char *),
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dgbcon_(char *, integer *, integer *, integer *, doublereal *,
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integer *, integer *, doublereal *, doublereal *, doublereal *,
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integer *, integer *);
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static doublereal colcnd;
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extern doublereal dlantb_(char *, char *, char *, integer *, integer *,
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doublereal *, integer *, doublereal *);
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extern /* Subroutine */ int dgbequ_(integer *, integer *, integer *,
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integer *, doublereal *, integer *, doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, integer *), dgbrfs_(
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char *, integer *, integer *, integer *, integer *, doublereal *,
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integer *, doublereal *, integer *, integer *, doublereal *,
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integer *, doublereal *, integer *, doublereal *, doublereal *,
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doublereal *, integer *, integer *), dgbtrf_(integer *,
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integer *, integer *, integer *, doublereal *, integer *, integer
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*, integer *);
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static logical nofact;
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extern /* Subroutine */ int dlacpy_(char *, integer *, integer *,
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doublereal *, integer *, doublereal *, integer *),
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xerbla_(char *, integer *);
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static doublereal bignum;
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extern /* Subroutine */ int dgbtrs_(char *, integer *, integer *, integer
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*, integer *, doublereal *, integer *, integer *, doublereal *,
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integer *, integer *);
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static integer infequ;
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static logical colequ;
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static doublereal rowcnd;
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static logical notran;
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static doublereal smlnum;
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static logical rowequ;
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static doublereal rpvgrw;
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#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
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#define x_ref(a_1,a_2) x[(a_2)*x_dim1 + a_1]
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#define ab_ref(a_1,a_2) ab[(a_2)*ab_dim1 + a_1]
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#define afb_ref(a_1,a_2) afb[(a_2)*afb_dim1 + a_1]
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ab_dim1 = *ldab;
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ab_offset = 1 + ab_dim1 * 1;
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ab -= ab_offset;
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afb_dim1 = *ldafb;
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afb_offset = 1 + afb_dim1 * 1;
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afb -= afb_offset;
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--ipiv;
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--r__;
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--c__;
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b_dim1 = *ldb;
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b_offset = 1 + b_dim1 * 1;
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b -= b_offset;
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x_dim1 = *ldx;
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x_offset = 1 + x_dim1 * 1;
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x -= x_offset;
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--ferr;
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--berr;
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--work;
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--iwork;
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/* Function Body */
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*info = 0;
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nofact = lsame_(fact, "N");
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equil = lsame_(fact, "E");
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notran = lsame_(trans, "N");
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if (nofact || equil) {
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*(unsigned char *)equed = 'N';
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rowequ = FALSE_;
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colequ = FALSE_;
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} else {
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rowequ = lsame_(equed, "R") || lsame_(equed,
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"B");
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colequ = lsame_(equed, "C") || lsame_(equed,
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"B");
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smlnum = dlamch_("Safe minimum");
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bignum = 1. / smlnum;
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}
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/* Test the input parameters. */
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if (! nofact && ! equil && ! lsame_(fact, "F")) {
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*info = -1;
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} else if (! notran && ! lsame_(trans, "T") && !
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lsame_(trans, "C")) {
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*info = -2;
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} else if (*n < 0) {
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*info = -3;
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} else if (*kl < 0) {
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*info = -4;
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} else if (*ku < 0) {
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*info = -5;
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} else if (*nrhs < 0) {
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*info = -6;
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} else if (*ldab < *kl + *ku + 1) {
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*info = -8;
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} else if (*ldafb < (*kl << 1) + *ku + 1) {
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*info = -10;
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} else if (lsame_(fact, "F") && ! (rowequ || colequ
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|| lsame_(equed, "N"))) {
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*info = -12;
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} else {
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if (rowequ) {
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rcmin = bignum;
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rcmax = 0.;
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i__1 = *n;
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for (j = 1; j <= i__1; ++j) {
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/* Computing MIN */
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d__1 = rcmin, d__2 = r__[j];
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rcmin = min(d__1,d__2);
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/* Computing MAX */
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d__1 = rcmax, d__2 = r__[j];
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rcmax = max(d__1,d__2);
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/* L10: */
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}
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if (rcmin <= 0.) {
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*info = -13;
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} else if (*n > 0) {
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rowcnd = max(rcmin,smlnum) / min(rcmax,bignum);
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} else {
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rowcnd = 1.;
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}
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}
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if (colequ && *info == 0) {
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rcmin = bignum;
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rcmax = 0.;
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i__1 = *n;
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for (j = 1; j <= i__1; ++j) {
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/* Computing MIN */
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d__1 = rcmin, d__2 = c__[j];
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rcmin = min(d__1,d__2);
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/* Computing MAX */
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d__1 = rcmax, d__2 = c__[j];
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rcmax = max(d__1,d__2);
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/* L20: */
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}
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if (rcmin <= 0.) {
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*info = -14;
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} else if (*n > 0) {
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colcnd = max(rcmin,smlnum) / min(rcmax,bignum);
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} else {
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colcnd = 1.;
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}
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}
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if (*info == 0) {
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if (*ldb < max(1,*n)) {
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*info = -16;
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} else if (*ldx < max(1,*n)) {
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*info = -18;
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}
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}
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}
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if (*info != 0) {
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i__1 = -(*info);
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xerbla_("DGBSVX", &i__1);
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return 0;
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}
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if (equil) {
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/* Compute row and column scalings to equilibrate the matrix A. */
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dgbequ_(n, n, kl, ku, &ab[ab_offset], ldab, &r__[1], &c__[1], &rowcnd,
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&colcnd, &amax, &infequ);
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if (infequ == 0) {
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/* Equilibrate the matrix. */
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dlaqgb_(n, n, kl, ku, &ab[ab_offset], ldab, &r__[1], &c__[1], &
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|
rowcnd, &colcnd, &amax, equed);
|
|
rowequ = lsame_(equed, "R") || lsame_(equed,
|
|
"B");
|
|
colequ = lsame_(equed, "C") || lsame_(equed,
|
|
"B");
|
|
}
|
|
}
|
|
|
|
/* Scale the right hand side. */
|
|
|
|
if (notran) {
|
|
if (rowequ) {
|
|
i__1 = *nrhs;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
i__2 = *n;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
b_ref(i__, j) = r__[i__] * b_ref(i__, j);
|
|
/* L30: */
|
|
}
|
|
/* L40: */
|
|
}
|
|
}
|
|
} else if (colequ) {
|
|
i__1 = *nrhs;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
i__2 = *n;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
b_ref(i__, j) = c__[i__] * b_ref(i__, j);
|
|
/* L50: */
|
|
}
|
|
/* L60: */
|
|
}
|
|
}
|
|
|
|
if (nofact || equil) {
|
|
|
|
/* Compute the LU factorization of the band matrix A. */
|
|
|
|
i__1 = *n;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
/* Computing MAX */
|
|
i__2 = j - *ku;
|
|
j1 = max(i__2,1);
|
|
/* Computing MIN */
|
|
i__2 = j + *kl;
|
|
j2 = min(i__2,*n);
|
|
i__2 = j2 - j1 + 1;
|
|
dcopy_(&i__2, &ab_ref(*ku + 1 - j + j1, j), &c__1, &afb_ref(*kl +
|
|
*ku + 1 - j + j1, j), &c__1);
|
|
/* L70: */
|
|
}
|
|
|
|
dgbtrf_(n, n, kl, ku, &afb[afb_offset], ldafb, &ipiv[1], info);
|
|
|
|
/* Return if INFO is non-zero. */
|
|
|
|
if (*info != 0) {
|
|
if (*info > 0) {
|
|
|
|
/* Compute the reciprocal pivot growth factor of the
|
|
leading rank-deficient INFO columns of A. */
|
|
|
|
anorm = 0.;
|
|
i__1 = *info;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
/* Computing MAX */
|
|
i__2 = *ku + 2 - j;
|
|
/* Computing MIN */
|
|
i__4 = *n + *ku + 1 - j, i__5 = *kl + *ku + 1;
|
|
i__3 = min(i__4,i__5);
|
|
for (i__ = max(i__2,1); i__ <= i__3; ++i__) {
|
|
/* Computing MAX */
|
|
d__2 = anorm, d__3 = (d__1 = ab_ref(i__, j), abs(d__1)
|
|
);
|
|
anorm = max(d__2,d__3);
|
|
/* L80: */
|
|
}
|
|
/* L90: */
|
|
}
|
|
/* Computing MAX */
|
|
i__1 = 1, i__3 = *kl + *ku + 2 - *info;
|
|
/* Computing MIN */
|
|
i__4 = *info - 1, i__5 = *kl + *ku;
|
|
i__2 = min(i__4,i__5);
|
|
rpvgrw = dlantb_("M", "U", "N", info, &i__2, &afb_ref(max(
|
|
i__1,i__3), 1), ldafb, &work[1]);
|
|
if (rpvgrw == 0.) {
|
|
rpvgrw = 1.;
|
|
} else {
|
|
rpvgrw = anorm / rpvgrw;
|
|
}
|
|
work[1] = rpvgrw;
|
|
*rcond = 0.;
|
|
}
|
|
return 0;
|
|
}
|
|
}
|
|
|
|
/* Compute the norm of the matrix A and the
|
|
reciprocal pivot growth factor RPVGRW. */
|
|
|
|
if (notran) {
|
|
*(unsigned char *)norm = '1';
|
|
} else {
|
|
*(unsigned char *)norm = 'I';
|
|
}
|
|
anorm = dlangb_(norm, n, kl, ku, &ab[ab_offset], ldab, &work[1]);
|
|
i__1 = *kl + *ku;
|
|
rpvgrw = dlantb_("M", "U", "N", n, &i__1, &afb[afb_offset], ldafb, &work[
|
|
1]);
|
|
if (rpvgrw == 0.) {
|
|
rpvgrw = 1.;
|
|
} else {
|
|
rpvgrw = dlangb_("M", n, kl, ku, &ab[ab_offset], ldab, &work[1]) / rpvgrw;
|
|
}
|
|
|
|
/* Compute the reciprocal of the condition number of A. */
|
|
|
|
dgbcon_(norm, n, kl, ku, &afb[afb_offset], ldafb, &ipiv[1], &anorm, rcond,
|
|
&work[1], &iwork[1], info);
|
|
|
|
/* Set INFO = N+1 if the matrix is singular to working precision. */
|
|
|
|
if (*rcond < dlamch_("Epsilon")) {
|
|
*info = *n + 1;
|
|
}
|
|
|
|
/* Compute the solution matrix X. */
|
|
|
|
dlacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
|
|
dgbtrs_(trans, n, kl, ku, nrhs, &afb[afb_offset], ldafb, &ipiv[1], &x[
|
|
x_offset], ldx, info);
|
|
|
|
/* Use iterative refinement to improve the computed solution and
|
|
compute error bounds and backward error estimates for it. */
|
|
|
|
dgbrfs_(trans, n, kl, ku, nrhs, &ab[ab_offset], ldab, &afb[afb_offset],
|
|
ldafb, &ipiv[1], &b[b_offset], ldb, &x[x_offset], ldx, &ferr[1], &
|
|
berr[1], &work[1], &iwork[1], info);
|
|
|
|
/* Transform the solution matrix X to a solution of the original
|
|
system. */
|
|
|
|
if (notran) {
|
|
if (colequ) {
|
|
i__1 = *nrhs;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
i__3 = *n;
|
|
for (i__ = 1; i__ <= i__3; ++i__) {
|
|
x_ref(i__, j) = c__[i__] * x_ref(i__, j);
|
|
/* L100: */
|
|
}
|
|
/* L110: */
|
|
}
|
|
i__1 = *nrhs;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
ferr[j] /= colcnd;
|
|
/* L120: */
|
|
}
|
|
}
|
|
} else if (rowequ) {
|
|
i__1 = *nrhs;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
i__3 = *n;
|
|
for (i__ = 1; i__ <= i__3; ++i__) {
|
|
x_ref(i__, j) = r__[i__] * x_ref(i__, j);
|
|
/* L130: */
|
|
}
|
|
/* L140: */
|
|
}
|
|
i__1 = *nrhs;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
ferr[j] /= rowcnd;
|
|
/* L150: */
|
|
}
|
|
}
|
|
|
|
work[1] = rpvgrw;
|
|
return 0;
|
|
|
|
/* End of DGBSVX */
|
|
|
|
} /* dgbsvx_ */
|
|
|
|
#undef afb_ref
|
|
#undef ab_ref
|
|
#undef x_ref
|
|
#undef b_ref
|
|
|
|
|
|
#ifdef _cpluscplus
|
|
}
|
|
#endif
|