1ZPBSV(1) LAPACK driver routine (version 3.1) ZPBSV(1)
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6 ZPBSV - the solution to a complex system of linear equations A * X =
7 B,
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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, * )
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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
24 A = U**H * U, if UPLO = 'U', or
25 A = L * L**H, if UPLO = 'L',
26 where U is an upper triangular band matrix, and L is a lower triangular
27 band matrix, with the same number of superdiagonals or subdiagonals as
28 A. The factored form of A is then used to solve the system of equa‐
29 tions A * X = B.
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33 UPLO (input) CHARACTER*1
34 = 'U': Upper triangle of A is stored;
35 = 'L': Lower triangle of A is stored.
36 N (input) INTEGER
38 The number of linear equations, i.e., the order of the matrix
39 A. N >= 0.
40 KD (input) INTEGER
42 The number of superdiagonals of the matrix A if UPLO = 'U', or
43 the number of subdiagonals if UPLO = 'L'. KD >= 0.
44 NRHS (input) INTEGER
46 The number of right hand sides, i.e., the number of columns of
47 the matrix B. NRHS >= 0.
48 AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
50 On entry, the upper or lower triangle of the Hermitian band
51 matrix A, stored in the first KD+1 rows of the array. The j-th
52 column of A is stored in the j-th column of the array AB as
53 follows: if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-
54 KD)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
55 j<=i<=min(N,j+KD). See below for further details.
56 On exit, if INFO = 0, the triangular factor U or L from the
58 Cholesky factorization A = U**H*U or A = L*L**H of the band
59 matrix A, in the same storage format as A.
60 LDAB (input) INTEGER
62 The leading dimension of the array AB. LDAB >= KD+1.
63 B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
65 On entry, the N-by-NRHS right hand side matrix B. On exit, if
66 INFO = 0, the N-by-NRHS solution matrix X.
67 LDB (input) INTEGER
69 The leading dimension of the array B. LDB >= max(1,N).
70 INFO (output) INTEGER
72 = 0: successful exit
73 < 0: if INFO = -i, the i-th argument had an illegal value
74 > 0: if INFO = i, the leading minor of order i of A is not
75 positive definite, so the factorization could not be completed,
76 and the solution has not been computed.
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79 The band storage scheme is illustrated by the following example, when N
80 = 6, KD = 2, and UPLO = 'U':
81 On entry: On exit:
83 * * a13 a24 a35 a46 * * u13 u24 u35 u46
85 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
86 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
87 Similarly, if UPLO = 'L' the format of A is as follows:
89 On entry: On exit:
91 a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
93 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
94 a31 a42 a53 a64 * * l31 l42 l53 l64 * *
95 Array elements marked * are not used by the routine.
97 LAPACK driver routine (version 3.N1o)vember 2006 ZPBSV(1)