ZGBRFS(3F) ZGBRFS(3F)
ZGBRFS - improve the computed solution to a system of linear equations
when the coefficient matrix is banded, and provides error bounds and
backward error estimates for the solution
SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B,
LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
CHARACTER TRANS
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( *
), X( LDX, * )
ZGBRFS improves the computed solution to a system of linear equations
when the coefficient matrix is banded, and provides error bounds and
backward error estimates for the solution.
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 (Conjugate transpose)
N (input) INTEGER
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) COMPLEX*16 array, dimension (LDAB,N)
The original band matrix A, stored in rows 1 to KL+KU+1. The jth
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,jku)<=i<=min(n,j+kl).
Page 1
ZGBRFS(3F) ZGBRFS(3F)
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB (input) COMPLEX*16 array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as computed
by ZGBTRF. 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.
LDAFB (input) INTEGER
The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
IPIV (input) INTEGER array, dimension (N)
The pivot indices from ZGBTRF; for 1<=i<=N, row i of the matrix
was interchanged with row IPIV(i).
B (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by ZGBTRS. 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 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) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Page 2
ZGBRFS(3F) ZGBRFS(3F)
ITMAX is the maximum number of steps of iterative refinement.
ZGBRFS(3F) ZGBRFS(3F)
ZGBRFS - improve the computed solution to a system of linear equations
when the coefficient matrix is banded, and provides error bounds and
backward error estimates for the solution
SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B,
LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
CHARACTER TRANS
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( *
), X( LDX, * )
ZGBRFS improves the computed solution to a system of linear equations
when the coefficient matrix is banded, and provides error bounds and
backward error estimates for the solution.
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 (Conjugate transpose)
N (input) INTEGER
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) COMPLEX*16 array, dimension (LDAB,N)
The original band matrix A, stored in rows 1 to KL+KU+1. The jth
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,jku)<=i<=min(n,j+kl).
Page 1
ZGBRFS(3F) ZGBRFS(3F)
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB (input) COMPLEX*16 array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as computed
by ZGBTRF. 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.
LDAFB (input) INTEGER
The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
IPIV (input) INTEGER array, dimension (N)
The pivot indices from ZGBTRF; for 1<=i<=N, row i of the matrix
was interchanged with row IPIV(i).
B (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by ZGBTRS. 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 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) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Page 2
ZGBRFS(3F) ZGBRFS(3F)
ITMAX is the maximum number of steps of iterative refinement.
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