1ZGBRFS(1) LAPACK routine (version 3.1) ZGBRFS(1)
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6 ZGBRFS - the computed solution to a system of linear equations when the
7 coefficient matrix is banded, and provides error bounds and backward
8 error estimates for the solution
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11 SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
12 B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
13 CHARACTER TRANS
15 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
17 INTEGER IPIV( * )
19 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
21 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( *
23 ), X( LDX, * )
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26 ZGBRFS improves the computed solution to a system of linear equations
27 when the coefficient matrix is banded, and provides error bounds and
28 backward error estimates for the solution.
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32 TRANS (input) CHARACTER*1
33 Specifies the form of the system of equations:
34 = 'N': A * X = B (No transpose)
35 = 'T': A**T * X = B (Transpose)
36 = 'C': A**H * X = B (Conjugate transpose)
37 N (input) INTEGER
39 The order of the matrix A. N >= 0.
40 KL (input) INTEGER
42 The number of subdiagonals within the band of A. KL >= 0.
43 KU (input) INTEGER
45 The number of superdiagonals within the band of A. KU >= 0.
46 NRHS (input) INTEGER
48 The number of right hand sides, i.e., the number of columns of
49 the matrices B and X. NRHS >= 0.
50 AB (input) COMPLEX*16 array, dimension (LDAB,N)
52 The original band matrix A, stored in rows 1 to KL+KU+1. The
53 j-th column of A is stored in the j-th column of the array AB
54 as follows: AB(ku+1+i-j,j) = A(i,j) for max(1,j-
55 ku)<=i<=min(n,j+kl).
56 LDAB (input) INTEGER
58 The leading dimension of the array AB. LDAB >= KL+KU+1.
59 AFB (input) COMPLEX*16 array, dimension (LDAFB,N)
61 Details of the LU factorization of the band matrix A, as com‐
62 puted by ZGBTRF. U is stored as an upper triangular band
63 matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the
64 multipliers used during the factorization are stored in rows
65 KL+KU+2 to 2*KL+KU+1.
66 LDAFB (input) INTEGER
68 The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
69 IPIV (input) INTEGER array, dimension (N)
71 The pivot indices from ZGBTRF; for 1<=i<=N, row i of the matrix
72 was interchanged with row IPIV(i).
73 B (input) COMPLEX*16 array, dimension (LDB,NRHS)
75 The right hand side matrix B.
76 LDB (input) INTEGER
78 The leading dimension of the array B. LDB >= max(1,N).
79 X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
81 On entry, the solution matrix X, as computed by ZGBTRS. On
82 exit, the improved solution matrix X.
83 LDX (input) INTEGER
85 The leading dimension of the array X. LDX >= max(1,N).
86 FERR (output) DOUBLE PRECISION array, dimension (NRHS)
88 The estimated forward error bound for each solution vector X(j)
89 (the j-th column of the solution matrix X). If XTRUE is the
90 true solution corresponding to X(j), FERR(j) is an estimated
91 upper bound for the magnitude of the largest element in (X(j) -
92 XTRUE) divided by the magnitude of the largest element in X(j).
93 The estimate is as reliable as the estimate for RCOND, and is
94 almost always a slight overestimate of the true error.
95 BERR (output) DOUBLE PRECISION array, dimension (NRHS)
97 The componentwise relative backward error of each solution vec‐
98 tor X(j) (i.e., the smallest relative change in any element of
99 A or B that makes X(j) an exact solution).
100 WORK (workspace) COMPLEX*16 array, dimension (2*N)
102 RWORK (workspace) DOUBLE PRECISION array, dimension (N)
104 INFO (output) INTEGER
106 = 0: successful exit
107 < 0: if INFO = -i, the i-th argument had an illegal value
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110 ITMAX is the maximum number of steps of iterative refinement.
111 LAPACK routine (version 3.1) November 2006 ZGBRFS(1)