1CPBSTF(1) LAPACK routine (version 3.2) CPBSTF(1)
2
6 CPBSTF - computes a split Cholesky factorization of a complex Hermitian
7 positive definite band matrix A
8
10 SUBROUTINE CPBSTF( UPLO, N, KD, AB, LDAB, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, KD, LDAB, N
15 COMPLEX AB( LDAB, * )
17
19 CPBSTF computes a split Cholesky factorization of a complex Hermitian
20 positive definite band matrix A. This routine is designed to be used
21 in conjunction with CHBGST. The factorization has the form A = S**H*S
22 where S is a band matrix of the same bandwidth as A and the following
23 structure:
24 S = ( U )
25 ( M L )
26 where U is upper triangular of order m = (n+kd)/2, and L is lower tri‐
27 angular of order n-m.
28
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 AB (input/output) COMPLEX array, dimension (LDAB,N)
42 On entry, the upper or lower triangle of the Hermitian band
43 matrix A, stored in the first kd+1 rows of the array. The j-th
44 column of A is stored in the j-th column of the array AB as
45 follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-
46 kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
47 j<=i<=min(n,j+kd). On exit, if INFO = 0, the factor S from the
48 split Cholesky factorization A = S**H*S. See Further Details.
49 LDAB (input) INTEGER The leading dimension of the array AB.
50 LDAB >= KD+1.
51 INFO (output) INTEGER
53 = 0: successful exit
54 < 0: if INFO = -i, the i-th argument had an illegal value
55 > 0: if INFO = i, the factorization could not be completed,
56 because the updated element a(i,i) was negative; the matrix A
57 is not positive definite.
58
60 The band storage scheme is illustrated by the following example, when N
61 = 7, KD = 2:
62 S = ( s11 s12 s13 )
63 ( s22 s23 s24 )
64 ( s33 s34 )
65 ( s44 )
66 ( s53 s54 s55 )
67 ( s64 s65 s66 )
68 ( s75 s76 s77 )
69 If UPLO = 'U', the array AB holds:
70 on entry: on exit:
71 * * a13 a24 a35 a46 a57 * * s13 s24 s53' s64' s75'
72 * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54' s65' s76'
73 a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 If
74 UPLO = 'L', the array AB holds:
75 on entry: on exit:
76 a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
77 a21 a32 a43 a54 a65 a76 * s12' s23' s34' s54 s65 s76 * a31
78 a42 a53 a64 a64 * * s13' s24' s53 s64 s75 * * Array
79 elements marked * are not used by the routine; s12' denotes conjg(s12);
80 the diagonal elements of S are real.
81 LAPACK routine (version 3.2) November 2008 CPBSTF(1)