sgbsvx(3)

NAME

SGBSVX - use 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 subdi

agonals and KU superdiagonals, and X and B are N-by-NRHS matri

ces.

Error bounds on the solution and a condition estimate are

also provided.

DESCRIPTION

The following steps are performed by this subroutine:

1. If FACT = 'E', real scaling factors are computed to

equilibrate
the system:
TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X

= diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X

= diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X

= diag(C)*B

Whether or not the system will be equilibrated depends

on the
scaling of the matrix A, but if equilibration is used,

A is
overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if

TRANS='N')
or diag(C)*B (if TRANS = 'T' or 'C').

2. If FACT = 'N' or 'E', the LU decomposition is used to

factor the
matrix A (after equilibration if FACT = 'E') as
A = L * U,

where L is a product of permutation and unit lower tri

angular
matrices with KL subdiagonals, and U is upper triangu

lar with
KL+KU superdiagonals.

3. If some U(i,i)=0, so that U is exactly singular, then

the routine
returns with INFO = i. Otherwise, the factored form of

A is used
to estimate the condition number of the matrix A. If

the
reciprocal of the condition number is less than machine

precision,
INFO = N+1 is returned as a warning, but the routine

still goes on
to solve for X and compute error bounds as described

below.

4. The system of equations is solved for X using the fac

tored 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 premulti

plied by
diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or

'C') so
that it solves the original system before equilibra

tion.

ARGUMENTS

FACT (input) CHARACTER*1

Specifies whether or not the factored form of the

matrix A is supplied on entry, and if not, whether the matrix A

should be equilibrated before it is factored. = 'F': On entry,

AFB and IPIV contain the factored form of A. If EQUED is not

'N', the matrix A has been equilibrated with scaling factors giv

en by R and C. AB, AFB, and IPIV are not modified. = 'N': The

matrix A will be copied to AFB and factored.
= 'E': The matrix A will be equilibrated if nec

essary, then copied to AFB and factored.

TRANS (input) CHARACTER*1
Specifies the form of the system of equations. =

'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': 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 j-th column of A is stored in the j-th column of

the array AB as follows: AB(KU+1+i-j,j) = A(i,j) for max(1,j

KU)<=i<=min(N,j+kl)

If FACT = 'F' and EQUED is not 'N', then A must

have been 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 fol

lows: 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) REAL 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 ma

trix A, as computed by SGBTRF. U is stored as an upper triangu

lar 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 equili

brated 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 SGBTRF; 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 pre

multiplied 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 * di

ag(C). EQUED is an input argument if FACT = 'F'; otherwise, it

is an output argument.

R (input or output) REAL 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) REAL 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) REAL 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) REAL 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) 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 ma

trix is singular to working precision. This condition is indi

cated by a return code of INFO > 0.

FERR (output) REAL array, dimension (NRHS)The estimated forward error bound for each solu

tion 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) 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 factoriza

tion 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 ille

gal 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 re

turned. = N+1: U is nonsingular, but RCOND is less than machine

precision, meaning that the matrix is singular to working preci

sion. Nevertheless, the solution and error bounds are computed

because there are a number of situations where the computed solu

tion can be more accurate than the value of RCOND would suggest.

LAPACK version 3.0 15 June 2000

SGBSVX(3)