1ZSPTRF(1) LAPACK routine (version 3.1) ZSPTRF(1)
2
6 ZSPTRF - the factorization of a complex symmetric matrix A stored in
7 packed format using the Bunch-Kaufman diagonal pivoting method
8
10 SUBROUTINE ZSPTRF( UPLO, N, AP, IPIV, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, N
15 INTEGER IPIV( * )
17 COMPLEX*16 AP( * )
19
21 ZSPTRF computes the factorization of a complex symmetric matrix A
22 stored in packed format using the Bunch-Kaufman diagonal pivoting
23 method:
24 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‐
28 angular matrices, and D is symmetric and block diagonal with 1-by-1 and
29 2-by-2 diagonal blocks.
30
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 order of the matrix A. N >= 0.
39 AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
41 On entry, the upper or lower triangle of the symmetric matrix
42 A, packed columnwise in a linear array. The j-th column of A
43 is stored in the array AP as follows: if UPLO = 'U', AP(i +
44 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
45 (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
46 On exit, the block diagonal matrix D and the multipliers used
48 to obtain the factor U or L, stored as a packed triangular
49 matrix overwriting A (see below for further details).
50 IPIV (output) INTEGER array, dimension (N)
52 Details of the interchanges and the block structure of D. If
53 IPIV(k) > 0, then rows and columns k and IPIV(k) were inter‐
54 changed and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U'
55 and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
56 -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diag‐
57 onal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then
58 rows and columns k+1 and -IPIV(k) were interchanged and
59 D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
60 INFO (output) INTEGER
62 = 0: successful exit
63 < 0: if INFO = -i, the i-th argument had an illegal value
64 > 0: if INFO = i, D(i,i) is exactly zero. The factorization
65 has been completed, but the block diagonal matrix D is exactly
66 singular, and division by zero will occur if it is used to
67 solve a system of equations.
68
70 5-96 - Based on modifications by J. Lewis, Boeing Computer Services
71 Company
72 If UPLO = 'U', then A = U*D*U', where
74 U = P(n)*U(n)* ... *P(k)U(k)* ...,
75 i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1
76 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
77 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
78 by IPIV(k), and U(k) is a unit upper triangular matrix, such that if
79 the diagonal block D(k) is of order s (s = 1 or 2), then
80 ( I v 0 ) k-s
82 U(k) = ( 0 I 0 ) s
83 ( 0 0 I ) n-k
84 k-s s n-k
85 If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s =
87 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and
88 A(k,k), and v overwrites A(1:k-2,k-1:k).
89 If UPLO = 'L', then A = L*D*L', where
91 L = P(1)*L(1)* ... *P(k)*L(k)* ...,
92 i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n
93 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
94 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
95 by IPIV(k), and L(k) is a unit lower triangular matrix, such that if
96 the diagonal block D(k) is of order s (s = 1 or 2), then
97 ( I 0 0 ) k-1
99 L(k) = ( 0 I 0 ) s
100 ( 0 v I ) n-k-s+1
101 k-1 s n-k-s+1
102 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s =
104 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and
105 A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
106 LAPACK routine (version 3.1) November 2006 ZSPTRF(1)