1ZPBSV(1) LAPACK driver routine (version 3.2) ZPBSV(1)
2
6 ZPBSV - computes the solution to a complex system of linear equations
7 A * X = B,
8
10 SUBROUTINE ZPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, KD, LDAB, LDB, N, NRHS
15 COMPLEX*16 AB( LDAB, * ), B( LDB, * )
17
19 ZPBSV computes the solution to a complex system of linear equations
20 A * X = B, where A is an N-by-N Hermitian positive definite band
21 matrix and X and B are N-by-NRHS matrices.
22 The Cholesky decomposition is used to factor A as
23 A = U**H * U, if UPLO = 'U', or
24 A = L * L**H, if UPLO = 'L',
25 where U is an upper triangular band matrix, and L is a lower triangular
26 band matrix, with the same number of superdiagonals or subdiagonals as
27 A. The factored form of A is then used to solve the system of equa‐
28 tions A * X = B.
29
31 UPLO (input) CHARACTER*1
32 = 'U': Upper triangle of A is stored;
33 = 'L': Lower triangle of A is stored.
34 N (input) INTEGER
36 The number of linear equations, i.e., the order of the matrix
37 A. N >= 0.
38 KD (input) INTEGER
40 The number of superdiagonals of the matrix A if UPLO = 'U', or
41 the number of subdiagonals if UPLO = 'L'. KD >= 0.
42 NRHS (input) INTEGER
44 The number of right hand sides, i.e., the number of columns of
45 the matrix B. NRHS >= 0.
46 AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
48 On entry, the upper or lower triangle of the Hermitian band
49 matrix A, stored in the first KD+1 rows of the array. The j-th
50 column of A is stored in the j-th column of the array AB as
51 follows: if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-
52 KD)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
53 j<=i<=min(N,j+KD). See below for further details. On exit, if
54 INFO = 0, the triangular factor U or L from the Cholesky fac‐
55 torization A = U**H*U or A = L*L**H of the band matrix A, in
56 the same storage format as A.
57 LDAB (input) INTEGER
59 The leading dimension of the array AB. LDAB >= KD+1.
60 B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
62 On entry, the N-by-NRHS right hand side matrix B. On exit, if
63 INFO = 0, the N-by-NRHS solution matrix X.
64 LDB (input) INTEGER
66 The leading dimension of the array B. LDB >= max(1,N).
67 INFO (output) INTEGER
69 = 0: successful exit
70 < 0: if INFO = -i, the i-th argument had an illegal value
71 > 0: if INFO = i, the leading minor of order i of A is not
72 positive definite, so the factorization could not be completed,
73 and the solution has not been computed.
74
76 The band storage scheme is illustrated by the following example, when N
77 = 6, KD = 2, and UPLO = 'U':
78 On entry: On exit:
79 * * a13 a24 a35 a46 * * u13 u24 u35 u46
80 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
81 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 Simi‐
82 larly, if UPLO = 'L' the format of A is as follows:
83 On entry: On exit:
84 a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
85 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
86 a31 a42 a53 a64 * * l31 l42 l53 l64 * * Array
87 elements marked * are not used by the routine.
88 LAPACK driver routine (version 3.N2o)vember 2008 ZPBSV(1)