1CHPTRF(1) LAPACK routine (version 3.2) CHPTRF(1)
2
6 CHPTRF - computes the factorization of a complex Hermitian packed
7 matrix A using the Bunch-Kaufman diagonal pivoting method
8
10 SUBROUTINE CHPTRF( UPLO, N, AP, IPIV, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, N
15 INTEGER IPIV( * )
17 COMPLEX AP( * )
19
21 CHPTRF computes the factorization of a complex Hermitian packed matrix
22 A using the Bunch-Kaufman diagonal pivoting method:
23 A = U*D*U**H or A = L*D*L**H
24 where U (or L) is a product of permutation and unit upper (lower) tri‐
25 angular matrices, and D is Hermitian and block diagonal with 1-by-1 and
26 2-by-2 diagonal blocks.
27
29 UPLO (input) CHARACTER*1
30 = 'U': Upper triangle of A is stored;
31 = 'L': Lower triangle of A is stored.
32 N (input) INTEGER
34 The order of the matrix A. N >= 0.
35 AP (input/output) COMPLEX array, dimension (N*(N+1)/2)
37 On entry, the upper or lower triangle of the Hermitian matrix
38 A, packed columnwise in a linear array. The j-th column of A
39 is stored in the array AP as follows: if UPLO = 'U', AP(i +
40 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
41 (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. On exit, the block diag‐
42 onal matrix D and the multipliers used to obtain the factor U
43 or L, stored as a packed triangular matrix overwriting A (see
44 below for further details).
45 IPIV (output) INTEGER array, dimension (N)
47 Details of the interchanges and the block structure of D. If
48 IPIV(k) > 0, then rows and columns k and IPIV(k) were inter‐
49 changed and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U'
50 and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
51 -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diag‐
52 onal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then
53 rows and columns k+1 and -IPIV(k) were interchanged and
54 D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
55 INFO (output) INTEGER
57 = 0: successful exit
58 < 0: if INFO = -i, the i-th argument had an illegal value
59 > 0: if INFO = i, D(i,i) is exactly zero. The factorization
60 has been completed, but the block diagonal matrix D is exactly
61 singular, and division by zero will occur if it is used to
62 solve a system of equations.
63
65 5-96 - Based on modifications by J. Lewis, Boeing Computer Services
66 Company
67 If UPLO = 'U', then A = U*D*U', where
68 U = P(n)*U(n)* ... *P(k)U(k)* ...,
69 i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1
70 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
71 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
72 by IPIV(k), and U(k) is a unit upper triangular matrix, such that if
73 the diagonal block D(k) is of order s (s = 1 or 2), then
74 ( I v 0 ) k-s
75 U(k) = ( 0 I 0 ) s
76 ( 0 0 I ) n-k
77 k-s s n-k
78 If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s =
79 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and
80 A(k,k), and v overwrites A(1:k-2,k-1:k).
81 If UPLO = 'L', then A = L*D*L', where
82 L = P(1)*L(1)* ... *P(k)*L(k)* ...,
83 i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n
84 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
85 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
86 by IPIV(k), and L(k) is a unit lower triangular matrix, such that if
87 the diagonal block D(k) is of order s (s = 1 or 2), then
88 ( I 0 0 ) k-1
89 L(k) = ( 0 I 0 ) s
90 ( 0 v I ) n-k-s+1
91 k-1 s n-k-s+1
92 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s =
93 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and
94 A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
95 LAPACK routine (version 3.2) November 2008 CHPTRF(1)