1SGBTRF(1)                LAPACK routine (version 3.1)                SGBTRF(1)
2
3
4

NAME

6       SGBTRF  - an LU factorization of a real m-by-n band matrix A using par‐
7       tial pivoting with row interchanges
8

SYNOPSIS

10       SUBROUTINE SGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
11
12           INTEGER        INFO, KL, KU, LDAB, M, N
13
14           INTEGER        IPIV( * )
15
16           REAL           AB( LDAB, * )
17

PURPOSE

19       SGBTRF computes an LU factorization of a  real  m-by-n  band  matrix  A
20       using partial pivoting with row interchanges.
21
22       This is the blocked version of the algorithm, calling Level 3 BLAS.
23
24

ARGUMENTS

26       M       (input) INTEGER
27               The number of rows of the matrix A.  M >= 0.
28
29       N       (input) INTEGER
30               The number of columns of the matrix A.  N >= 0.
31
32       KL      (input) INTEGER
33               The number of subdiagonals within the band of A.  KL >= 0.
34
35       KU      (input) INTEGER
36               The number of superdiagonals within the band of A.  KU >= 0.
37
38       AB      (input/output) REAL array, dimension (LDAB,N)
39               On  entry,  the  matrix  A  in  band  storage,  in rows KL+1 to
40               2*KL+KU+1; rows 1 to KL of the array need not be set.  The j-th
41               column  of  A  is  stored in the j-th column of the array AB as
42               follows:    AB(kl+ku+1+i-j,j)    =    A(i,j)    for    max(1,j-
43               ku)<=i<=min(m,j+kl)
44
45               On  exit, details of the factorization: U is stored as an upper
46               triangular band matrix with KL+KU superdiagonals in rows  1  to
47               KL+KU+1,  and the multipliers used during the factorization are
48               stored in rows KL+KU+2 to 2*KL+KU+1.   See  below  for  further
49               details.
50
51       LDAB    (input) INTEGER
52               The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
53
54       IPIV    (output) INTEGER array, dimension (min(M,N))
55               The  pivot indices; for 1 <= i <= min(M,N), row i of the matrix
56               was interchanged with row IPIV(i).
57
58       INFO    (output) INTEGER
59               = 0: successful exit
60               < 0: if INFO = -i, the i-th argument had an illegal value
61               > 0: if INFO = +i, U(i,i) is exactly  zero.  The  factorization
62               has  been  completed, but the factor U is exactly singular, and
63               division by zero will occur if it is used to solve a system  of
64               equations.
65

FURTHER DETAILS

67       The band storage scheme is illustrated by the following example, when M
68       = N = 6, KL = 2, KU = 1:
69
70       On entry:                       On exit:
71
72           *    *    *    +    +    +       *    *    *   u14  u25  u36
73           *    *    +    +    +    +       *    *   u13  u24  u35  u46
74           *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
75          a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
76          a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
77          a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
78
79       Array elements marked * are not used by the routine; elements marked  +
80       need not be set on entry, but are required by the routine to store ele‐
81       ments of U because of fill-in resulting from the row interchanges.
82
83
84
85
86 LAPACK routine (version 3.1)    November 2006                       SGBTRF(1)