1SPBRFS(1) LAPACK routine (version 3.2) SPBRFS(1)
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6 SPBRFS - improves the computed solution to a system of linear equations
7 when the coefficient matrix is symmetric positive definite and banded,
8 and provides error bounds and backward error estimates for the solution
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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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30 UPLO (input) CHARACTER*1
31 = 'U': Upper triangle of A is stored;
32 = 'L': Lower triangle of A is stored.
33 N (input) INTEGER
35 The order of the matrix A. N >= 0.
36 KD (input) INTEGER
38 The number of superdiagonals of the matrix A if UPLO = 'U', or
39 the number of subdiagonals if UPLO = 'L'. KD >= 0.
40 NRHS (input) INTEGER
42 The number of right hand sides, i.e., the number of columns of
43 the matrices B and X. NRHS >= 0.
44 AB (input) REAL array, dimension (LDAB,N)
46 The upper or lower triangle of the symmetric band matrix A,
47 stored in the first KD+1 rows of the array. The j-th column of
48 A is stored in the j-th column of the array AB as follows: if
49 UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if
50 UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
51 LDAB (input) INTEGER
53 The leading dimension of the array AB. LDAB >= KD+1.
54 AFB (input) REAL array, dimension (LDAFB,N)
56 The triangular factor U or L from the Cholesky factorization A
57 = U**T*U or A = L*L**T of the band matrix A as computed by SPB‐
58 TRF, in the same storage format as A (see AB).
59 LDAFB (input) INTEGER
61 The leading dimension of the array AFB. LDAFB >= KD+1.
62 B (input) REAL array, dimension (LDB,NRHS)
64 The right hand side matrix B.
65 LDB (input) INTEGER
67 The leading dimension of the array B. LDB >= max(1,N).
68 X (input/output) REAL array, dimension (LDX,NRHS)
70 On entry, the solution matrix X, as computed by SPBTRS. On
71 exit, the improved solution matrix X.
72 LDX (input) INTEGER
74 The leading dimension of the array X. LDX >= max(1,N).
75 FERR (output) REAL array, dimension (NRHS)
77 The estimated forward error bound for each solution vector X(j)
78 (the j-th column of the solution matrix X). If XTRUE is the
79 true solution corresponding to X(j), FERR(j) is an estimated
80 upper bound for the magnitude of the largest element in (X(j) -
81 XTRUE) divided by the magnitude of the largest element in X(j).
82 The estimate is as reliable as the estimate for RCOND, and is
83 almost always a slight overestimate of the true error.
84 BERR (output) REAL array, dimension (NRHS)
86 The componentwise relative backward error of each solution vec‐
87 tor X(j) (i.e., the smallest relative change in any element of
88 A or B that makes X(j) an exact solution).
89 WORK (workspace) REAL array, dimension (3*N)
91 IWORK (workspace) INTEGER array, dimension (N)
93 INFO (output) INTEGER
95 = 0: successful exit
96 < 0: if INFO = -i, the i-th argument had an illegal value
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99 ITMAX is the maximum number of steps of iterative refinement.
100 LAPACK routine (version 3.2) November 2008 SPBRFS(1)