1SGBTRF(1) LAPACK routine (version 3.2) SGBTRF(1)
2
6 SGBTRF - computes an LU factorization of a real m-by-n band matrix A
7 using partial pivoting with row interchanges
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10 SUBROUTINE SGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
11 INTEGER INFO, KL, KU, LDAB, M, N
13 INTEGER IPIV( * )
15 REAL AB( LDAB, * )
17
19 SGBTRF computes an LU factorization of a real m-by-n band matrix A
20 using partial pivoting with row interchanges. This is the blocked ver‐
21 sion of the algorithm, calling Level 3 BLAS.
22
24 M (input) INTEGER
25 The number of rows of the matrix A. M >= 0.
26 N (input) INTEGER
28 The number of columns of the matrix A. N >= 0.
29 KL (input) INTEGER
31 The number of subdiagonals within the band of A. KL >= 0.
32 KU (input) INTEGER
34 The number of superdiagonals within the band of A. KU >= 0.
35 AB (input/output) REAL array, dimension (LDAB,N)
37 On entry, the matrix A in band storage, in rows KL+1 to
38 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th
39 column of A is stored in the j-th column of the array AB as
40 follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-
41 ku)<=i<=min(m,j+kl) On exit, details of the factorization: U is
42 stored as an upper triangular band matrix with KL+KU superdiag‐
43 onals in rows 1 to KL+KU+1, and the multipliers used during the
44 factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See
45 below for further details.
46 LDAB (input) INTEGER
48 The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
49 IPIV (output) INTEGER array, dimension (min(M,N))
51 The pivot indices; for 1 <= i <= min(M,N), row i of the matrix
52 was interchanged with row IPIV(i).
53 INFO (output) INTEGER
55 = 0: successful exit
56 < 0: if INFO = -i, the i-th argument had an illegal value
57 > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
58 has been completed, but the factor U is exactly singular, and
59 division by zero will occur if it is used to solve a system of
60 equations.
61
63 The band storage scheme is illustrated by the following example, when M
64 = N = 6, KL = 2, KU = 1:
65 On entry: On exit:
66 * * * + + + * * * u14 u25 u36
67 * * + + + + * * u13 u24 u35 u46
68 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
69 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
70 a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
71 a31 a42 a53 a64 * * m31 m42 m53 m64 * * Array
72 elements marked * are not used by the routine; elements marked + need
73 not be set on entry, but are required by the routine to store elements
74 of U because of fill-in resulting from the row interchanges.
75 LAPACK routine (version 3.2) November 2008 SGBTRF(1)