1SPBTRF(1) LAPACK routine (version 3.2) SPBTRF(1)
2
6 SPBTRF - computes the Cholesky factorization of a real symmetric posi‐
7 tive definite band matrix A
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10 SUBROUTINE SPBTRF( UPLO, N, KD, AB, LDAB, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, KD, LDAB, N
15 REAL AB( LDAB, * )
17
19 SPBTRF computes the Cholesky factorization of a real symmetric positive
20 definite band matrix A. The factorization has the form
21 A = U**T * U, if UPLO = 'U', or
22 A = L * L**T, if UPLO = 'L',
23 where U is an upper triangular matrix and L is lower triangular.
24
26 UPLO (input) CHARACTER*1
27 = 'U': Upper triangle of A is stored;
28 = 'L': Lower triangle of A is stored.
29 N (input) INTEGER
31 The order of the matrix A. N >= 0.
32 KD (input) INTEGER
34 The number of superdiagonals of the matrix A if UPLO = 'U', or
35 the number of subdiagonals if UPLO = 'L'. KD >= 0.
36 AB (input/output) REAL array, dimension (LDAB,N)
38 On entry, the upper or lower triangle of the symmetric band
39 matrix A, stored in the first KD+1 rows of the array. The j-th
40 column of A is stored in the j-th column of the array AB as
41 follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-
42 kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
43 j<=i<=min(n,j+kd). On exit, if INFO = 0, the triangular factor
44 U or L from the Cholesky factorization A = U**T*U or A = L*L**T
45 of the band matrix A, in the same storage format as A.
46 LDAB (input) INTEGER
48 The leading dimension of the array AB. LDAB >= KD+1.
49 INFO (output) INTEGER
51 = 0: successful exit
52 < 0: if INFO = -i, the i-th argument had an illegal value
53 > 0: if INFO = i, the leading minor of order i is not positive
54 definite, and the factorization could not be completed.
55
57 The band storage scheme is illustrated by the following example, when N
58 = 6, KD = 2, and UPLO = 'U':
59 On entry: On exit:
60 * * a13 a24 a35 a46 * * u13 u24 u35 u46
61 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
62 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 Simi‐
63 larly, if UPLO = 'L' the format of A is as follows:
64 On entry: On exit:
65 a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
66 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
67 a31 a42 a53 a64 * * l31 l42 l53 l64 * * Array
68 elements marked * are not used by the routine.
69 Contributed by
70 Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
71 LAPACK routine (version 3.2) November 2008 SPBTRF(1)