sgbsvx (l)  Linux Manuals
sgbsvx: uses the LU factorization to compute the solution to a real system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
NAME
SGBSVX  uses the LU factorization to compute the solution to a real system of linear equations A * X = B, A**T * X = B, or A**H * X = B,SYNOPSIS
 SUBROUTINE SGBSVX(
 FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
 CHARACTER EQUED, FACT, TRANS
 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
 REAL RCOND
 INTEGER IPIV( * ), IWORK( * )
 REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( * ), C( * ), FERR( * ), R( * ), WORK( * ), X( LDX, * )
PURPOSE
SGBSVX uses the LU factorization to compute the solution to a real system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are NbyNRHS matrices.Error bounds on the solution and a condition estimate are also provided.
DESCRIPTION
The following steps are performed by this subroutine:1. If FACT = aqEaq, real scaling factors are computed to equilibrate
the system:
TRANS
TRANS
TRANS
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(R)*A*diag(C)
or diag(C)*B
2. If FACT = aqNaq or aqEaq, the LU decomposition is used to factor the
matrix A
A
where L is a product of permutation and unit lower triangular
matrices with KL subdiagonals, and U is upper triangular with
KL+KU superdiagonals.
3. If some U(i,i)=0, so that U is exactly singular, then the routine
returns with INFO
to estimate the condition number of the matrix A.
reciprocal of the condition number is less than machine precision,
INFO
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(C)
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.
= aqFaq: On entry, AFB and IPIV contain the factored form of
A. If EQUED is not aqNaq, the matrix A has been
equilibrated with scaling factors given by R and C.
AB, AFB, and IPIV are not modified.
= aqNaq: The matrix A will be copied to AFB and factored.
= aqEaq: The matrix A will be equilibrated if necessary, then copied to AFB and factored.  TRANS (input) CHARACTER*1

Specifies the form of the system of equations.
= aqNaq: A * X = B (No transpose)
= aqTaq: A**T * X = B (Transpose)
= aqCaq: A**H * X = B (Transpose)  N (input) INTEGER
 The number of linear equations, i.e., the order of the matrix A. N >= 0.
 KL (input) INTEGER
 The number of subdiagonals within the band of A. KL >= 0.
 KU (input) INTEGER
 The number of superdiagonals within the band of A. KU >= 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) REAL array, dimension (LDAB,N)

On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
The jth column of A is stored in the jth column of the
array AB as follows:
AB(KU+1+ij,j) = A(i,j) for max(1,jKU)<=i<=min(N,j+kl)
If FACT = aqFaq and EQUED is not aqNaq, then A must have been
equilibrated by the scaling factors in R and/or C. AB is not
modified if FACT = aqFaq or aqNaq, or if FACT = aqEaq and
EQUED = aqNaq on exit.
On exit, if EQUED .ne. aqNaq, A is scaled as follows:
EQUED = aqRaq: A := diag(R) * A
EQUED = aqCaq: A := A * diag(C)
EQUED = aqBaq: A := diag(R) * A * diag(C).  LDAB (input) INTEGER
 The leading dimension of the array AB. LDAB >= KL+KU+1.
 AFB (input or output) REAL array, dimension (LDAFB,N)
 If FACT = aqFaq, then AFB is an input argument and on entry contains details of the LU factorization of the band matrix A, as computed by SGBTRF. 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. aqNaq, then AFB is the factored form of the equilibrated matrix A. If FACT = aqNaq, then AFB is an output argument and on exit returns details of the LU factorization of A. If FACT = aqEaq, 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 = aqFaq, then IPIV is an input argument and on entry contains the pivot indices from the factorization A = L*U as computed by SGBTRF; row i of the matrix was interchanged with row IPIV(i). If FACT = aqNaq, 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 = aqEaq, 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.
= aqNaq: No equilibration (always true if FACT = aqNaq).
= aqRaq: Row equilibration, i.e., A has been premultiplied by diag(R). = aqCaq: Column equilibration, i.e., A has been postmultiplied by diag(C). = aqBaq: Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C). EQUED is an input argument if FACT = aqFaq; otherwise, it is an output argument.  R (input or output) REAL array, dimension (N)
 The row scale factors for A. If EQUED = aqRaq or aqBaq, A is multiplied on the left by diag(R); if EQUED = aqNaq or aqCaq, R is not accessed. R is an input argument if FACT = aqFaq; otherwise, R is an output argument. If FACT = aqFaq and EQUED = aqRaq or aqBaq, each element of R must be positive.
 C (input or output) REAL array, dimension (N)
 The column scale factors for A. If EQUED = aqCaq or aqBaq, A is multiplied on the right by diag(C); if EQUED = aqNaq or aqRaq, C is not accessed. C is an input argument if FACT = aqFaq; otherwise, C is an output argument. If FACT = aqFaq and EQUED = aqCaq or aqBaq, each element of C must be positive.
 B (input/output) REAL array, dimension (LDB,NRHS)
 On entry, the right hand side matrix B. On exit, if EQUED = aqNaq, B is not modified; if TRANS = aqNaq and EQUED = aqRaq or aqBaq, B is overwritten by diag(R)*B; if TRANS = aqTaq or aqCaq and EQUED = aqCaq or aqBaq, B is overwritten by diag(C)*B.
 LDB (input) INTEGER
 The leading dimension of the array B. LDB >= max(1,N).
 X (output) REAL array, dimension (LDX,NRHS)
 If INFO = 0 or INFO = N+1, the NbyNRHS solution matrix X to the original system of equations. Note that A and B are modified on exit if EQUED .ne. aqNaq, and the solution to the equilibrated system is inv(diag(C))*X if TRANS = aqNaq and EQUED = aqCaq or aqBaq, or inv(diag(R))*X if TRANS = aqTaq or aqCaq and EQUED = aqRaq or aqBaq.
 LDX (input) INTEGER
 The leading dimension of the array X. LDX >= max(1,N).
 RCOND (output) REAL
 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) REAL array, dimension (NRHS)
 The estimated forward error bound for each solution vector X(j) (the jth 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) REAL array, dimension (NRHS)
 The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).
 WORK (workspace/output) REAL 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 ith 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.