DPBRFS - improve the computed solution to a system of lin
ear equations when the coefficient matrix is symmetric
positive definite and banded, and provides error bounds
and backward error estimates for the solution
SYNOPSIS
SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB,
LDAFB, B, LDB, X, LDX, FERR, BERR,
WORK, IWORK, INFO )
CHARACTER UPLO
INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N,
NRHS
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, *
), B( LDB, * ), BERR( * ), FERR( * ),
WORK( * ), X( LDX, * )
PURPOSE
DPBRFS improves the computed solution to a system of lin
ear equations when the coefficient matrix is symmetric
positive definite and banded, and provides error bounds
and backward error estimates for the solution.
ARGUMENTS
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if
UPLO = 'U', or the number of subdiagonals if UPLO
= 'L'. KD >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number
of columns of the matrices B and X. NRHS >= 0.
AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
The upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the
array. The j-th column of A is stored in the j-th
column of the array AB as follows: if UPLO = 'U',
AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if
UPLO = 'L', AB(1+i-j,j) = A(i,j) for
j<=i<=min(n,j+kd).
The leading dimension of the array AB. LDAB >=
KD+1.
AFB (input) DOUBLE PRECISION array, dimension
(LDAFB,N)
The triangular factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T of the band
matrix A as computed by DPBTRF, in the same stor
age format as A (see AB).
LDAFB (input) INTEGER
The leading dimension of the array AFB. LDAFB >=
KD+1.
B (input) DOUBLE PRECISION array, dimension
(LDB,NRHS)
The right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >=
max(1,N).
X (input/output) DOUBLE PRECISION array,
dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by
DPBTRS. On exit, the improved solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >=
max(1,N).
FERR (output) DOUBLE PRECISION 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 corre
sponding 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 reli
able as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each
solution vector X(j) (i.e., the smallest relative
change in any element of A or B that makes X(j) an
exact solution).
WORK (workspace) DOUBLE PRECISION array, dimension
(3*N)
IWORK (workspace) INTEGER array, dimension (N)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal value
PARAMETERS
ITMAX is the maximum number of steps of iterative refine
ment.
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