1ZSPTRF(1) LAPACK routine (version 3.2) ZSPTRF(1)
2
6 ZSPTRF - computes the factorization of a complex symmetric matrix A
7 stored in packed format using the Bunch-Kaufman diagonal pivoting
8 method
9
11 SUBROUTINE ZSPTRF( UPLO, N, AP, IPIV, INFO )
12 CHARACTER UPLO
14 INTEGER INFO, N
16 INTEGER IPIV( * )
18 COMPLEX*16 AP( * )
20
22 ZSPTRF computes the factorization of a complex symmetric matrix A
23 stored in packed format using the Bunch-Kaufman diagonal pivoting
24 method:
25 A = U*D*U**T or A = L*D*L**T
26 where U (or L) is a product of permutation and unit upper (lower) tri‐
27 angular matrices, and D is symmetric and block diagonal with 1-by-1 and
28 2-by-2 diagonal blocks.
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 order of the matrix A. N >= 0.
37 AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
39 On entry, the upper or lower triangle of the symmetric matrix
40 A, packed columnwise in a linear array. The j-th column of A
41 is stored in the array AP as follows: if UPLO = 'U', AP(i +
42 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
43 (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. On exit, the block diag‐
44 onal matrix D and the multipliers used to obtain the factor U
45 or L, stored as a packed triangular matrix overwriting A (see
46 below for further details).
47 IPIV (output) INTEGER array, dimension (N)
49 Details of the interchanges and the block structure of D. If
50 IPIV(k) > 0, then rows and columns k and IPIV(k) were inter‐
51 changed and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U'
52 and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
53 -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diag‐
54 onal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then
55 rows and columns k+1 and -IPIV(k) were interchanged and
56 D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
57 INFO (output) INTEGER
59 = 0: successful exit
60 < 0: if INFO = -i, the i-th argument had an illegal value
61 > 0: if INFO = i, D(i,i) is exactly zero. The factorization
62 has been completed, but the block diagonal matrix D is exactly
63 singular, and division by zero will occur if it is used to
64 solve a system of equations.
65
67 5-96 - Based on modifications by J. Lewis, Boeing Computer Services
68 Company
69 If UPLO = 'U', then A = U*D*U', where
70 U = P(n)*U(n)* ... *P(k)U(k)* ...,
71 i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1
72 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
73 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
74 by IPIV(k), and U(k) is a unit upper triangular matrix, such that if
75 the diagonal block D(k) is of order s (s = 1 or 2), then
76 ( I v 0 ) k-s
77 U(k) = ( 0 I 0 ) s
78 ( 0 0 I ) n-k
79 k-s s n-k
80 If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s =
81 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and
82 A(k,k), and v overwrites A(1:k-2,k-1:k).
83 If UPLO = 'L', then A = L*D*L', where
84 L = P(1)*L(1)* ... *P(k)*L(k)* ...,
85 i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n
86 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
87 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
88 by IPIV(k), and L(k) is a unit lower triangular matrix, such that if
89 the diagonal block D(k) is of order s (s = 1 or 2), then
90 ( I 0 0 ) k-1
91 L(k) = ( 0 I 0 ) s
92 ( 0 v I ) n-k-s+1
93 k-1 s n-k-s+1
94 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s =
95 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and
96 A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
97 LAPACK routine (version 3.2) November 2008 ZSPTRF(1)