1SPBRFS(1) LAPACK routine (version 3.1) SPBRFS(1)
2
6 SPBRFS - the computed solution to a system of linear equations when the
7 coefficient matrix is symmetric positive definite and banded, and pro‐
8 vides error bounds and backward error estimates for the solution
9
11 SUBROUTINE SPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
12 LDX, FERR, BERR, WORK, IWORK, INFO )
13 CHARACTER UPLO
15 INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
17 INTEGER IWORK( * )
19 REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( *
21 ), FERR( * ), WORK( * ), X( LDX, * )
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24 SPBRFS improves the computed solution to a system of linear equations
25 when the coefficient matrix is symmetric positive definite and banded,
26 and provides error bounds and backward error estimates for the solu‐
27 tion.
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31 UPLO (input) CHARACTER*1
32 = 'U': Upper triangle of A is stored;
33 = 'L': Lower triangle of A is stored.
34 N (input) INTEGER
36 The order of the matrix A. N >= 0.
37 KD (input) INTEGER
39 The number of superdiagonals of the matrix A if UPLO = 'U', or
40 the number of subdiagonals if UPLO = 'L'. KD >= 0.
41 NRHS (input) INTEGER
43 The number of right hand sides, i.e., the number of columns of
44 the matrices B and X. NRHS >= 0.
45 AB (input) REAL array, dimension (LDAB,N)
47 The upper or lower triangle of the symmetric band matrix A,
48 stored in the first KD+1 rows of the array. The j-th column of
49 A is stored in the j-th column of the array AB as follows: if
50 UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if
51 UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
52 LDAB (input) INTEGER
54 The leading dimension of the array AB. LDAB >= KD+1.
55 AFB (input) REAL array, dimension (LDAFB,N)
57 The triangular factor U or L from the Cholesky factorization A
58 = U**T*U or A = L*L**T of the band matrix A as computed by SPB‐
59 TRF, in the same storage format as A (see AB).
60 LDAFB (input) INTEGER
62 The leading dimension of the array AFB. LDAFB >= KD+1.
63 B (input) REAL array, dimension (LDB,NRHS)
65 The right hand side matrix B.
66 LDB (input) INTEGER
68 The leading dimension of the array B. LDB >= max(1,N).
69 X (input/output) REAL array, dimension (LDX,NRHS)
71 On entry, the solution matrix X, as computed by SPBTRS. On
72 exit, the improved solution matrix X.
73 LDX (input) INTEGER
75 The leading dimension of the array X. LDX >= max(1,N).
76 FERR (output) REAL array, dimension (NRHS)
78 The estimated forward error bound for each solution vector X(j)
79 (the j-th column of the solution matrix X). If XTRUE is the
80 true solution corresponding to X(j), FERR(j) is an estimated
81 upper bound for the magnitude of the largest element in (X(j) -
82 XTRUE) divided by the magnitude of the largest element in X(j).
83 The estimate is as reliable as the estimate for RCOND, and is
84 almost always a slight overestimate of the true error.
85 BERR (output) REAL array, dimension (NRHS)
87 The componentwise relative backward error of each solution vec‐
88 tor X(j) (i.e., the smallest relative change in any element of
89 A or B that makes X(j) an exact solution).
90 WORK (workspace) REAL array, dimension (3*N)
92 IWORK (workspace) INTEGER array, dimension (N)
94 INFO (output) INTEGER
96 = 0: successful exit
97 < 0: if INFO = -i, the i-th argument had an illegal value
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100 ITMAX is the maximum number of steps of iterative refinement.
101 LAPACK routine (version 3.1) November 2006 SPBRFS(1)