1SGBSV(1) LAPACK driver routine (version 3.1) SGBSV(1)
2
6 SGBSV - the solution to a real system of linear equations A * X = B,
7 where A is a band matrix of order N with KL subdiagonals and KU super‐
8 diagonals, and X and B are N-by-NRHS matrices
9
11 SUBROUTINE SGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )
12 INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS
14 INTEGER IPIV( * )
16 REAL AB( LDAB, * ), B( LDB, * )
18
20 SGBSV computes the solution to a real system of linear equations A * X
21 = B, where A is a band matrix of order N with KL subdiagonals and KU
22 superdiagonals, and X and B are N-by-NRHS matrices.
23 The LU decomposition with partial pivoting and row interchanges is used
25 to factor A as A = L * U, where L is a product of permutation and unit
26 lower triangular matrices with KL subdiagonals, and U is upper triangu‐
27 lar with KL+KU superdiagonals. The factored form of A is then used to
28 solve the system of equations A * X = B.
29
32 N (input) INTEGER
33 The number of linear equations, i.e., the order of the matrix
34 A. N >= 0.
35 KL (input) INTEGER
37 The number of subdiagonals within the band of A. KL >= 0.
38 KU (input) INTEGER
40 The number of superdiagonals within the band of A. KU >= 0.
41 NRHS (input) INTEGER
43 The number of right hand sides, i.e., the number of columns of
44 the matrix B. NRHS >= 0.
45 AB (input/output) REAL array, dimension (LDAB,N)
47 On entry, the matrix A in band storage, in rows KL+1 to
48 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th
49 column of A is stored in the j-th column of the array AB as
50 follows: AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-
51 KU)<=i<=min(N,j+KL) On exit, details of the factorization: U is
52 stored as an upper triangular band matrix with KL+KU superdiag‐
53 onals in rows 1 to KL+KU+1, and the multipliers used during the
54 factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See
55 below for further details.
56 LDAB (input) INTEGER
58 The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
59 IPIV (output) INTEGER array, dimension (N)
61 The pivot indices that define the permutation matrix P; row i
62 of the matrix was interchanged with row IPIV(i).
63 B (input/output) REAL array, dimension (LDB,NRHS)
65 On entry, the N-by-NRHS right hand side matrix B. On exit, if
66 INFO = 0, the N-by-NRHS solution matrix X.
67 LDB (input) INTEGER
69 The leading dimension of the array B. LDB >= max(1,N).
70 INFO (output) INTEGER
72 = 0: successful exit
73 < 0: if INFO = -i, the i-th argument had an illegal value
74 > 0: if INFO = i, U(i,i) is exactly zero. The factorization
75 has been completed, but the factor U is exactly singular, and
76 the solution has not been computed.
77
79 The band storage scheme is illustrated by the following example, when M
80 = N = 6, KL = 2, KU = 1:
81 On entry: On exit:
83 * * * + + + * * * u14 u25 u36
85 * * + + + + * * u13 u24 u35 u46
86 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
87 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
88 a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
89 a31 a42 a53 a64 * * m31 m42 m53 m64 * *
90 Array elements marked * are not used by the routine; elements marked +
92 need not be set on entry, but are required by the routine to store ele‐
93 ments of U because of fill-in resulting from the row interchanges.
94 LAPACK driver routine (version 3.N1o)vember 2006 SGBSV(1)