1DSPTRF(1) LAPACK routine (version 3.1) DSPTRF(1)
2
6 DSPTRF - the factorization of a real symmetric matrix A stored in
7 packed format using the Bunch-Kaufman diagonal pivoting method
8
10 SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, N
15 INTEGER IPIV( * )
17 DOUBLE PRECISION AP( * )
19
21 DSPTRF computes the factorization of a real symmetric matrix A stored
22 in packed format using the Bunch-Kaufman diagonal pivoting method:
23 A = U*D*U**T or A = L*D*L**T
25 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.
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32 UPLO (input) CHARACTER*1
33 = 'U': Upper triangle of A is stored;
34 = 'L': Lower triangle of A is stored.
35 N (input) INTEGER
37 The order of the matrix A. N >= 0.
38 AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
40 On entry, the upper or lower triangle of the symmetric matrix
41 A, packed columnwise in a linear array. The j-th column of A
42 is stored in the array AP as follows: if UPLO = 'U', AP(i +
43 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
44 (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
45 On exit, the block diagonal matrix D and the multipliers used
47 to obtain the factor U or L, stored as a packed triangular
48 matrix overwriting A (see below for further details).
49 IPIV (output) INTEGER array, dimension (N)
51 Details of the interchanges and the block structure of D. If
52 IPIV(k) > 0, then rows and columns k and IPIV(k) were inter‐
53 changed and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U'
54 and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
55 -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diag‐
56 onal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then
57 rows and columns k+1 and -IPIV(k) were interchanged and
58 D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
59 INFO (output) INTEGER
61 = 0: successful exit
62 < 0: if INFO = -i, the i-th argument had an illegal value
63 > 0: if INFO = i, D(i,i) is exactly zero. The factorization
64 has been completed, but the block diagonal matrix D is exactly
65 singular, and division by zero will occur if it is used to
66 solve a system of equations.
67
69 5-96 - Based on modifications by J. Lewis, Boeing Computer Services
70 Company
71 If UPLO = 'U', then A = U*D*U', where
73 U = P(n)*U(n)* ... *P(k)U(k)* ...,
74 i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1
75 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
76 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
77 by IPIV(k), and U(k) is a unit upper triangular matrix, such that if
78 the diagonal block D(k) is of order s (s = 1 or 2), then
79 ( I v 0 ) k-s
81 U(k) = ( 0 I 0 ) s
82 ( 0 0 I ) n-k
83 k-s s n-k
84 If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s =
86 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and
87 A(k,k), and v overwrites A(1:k-2,k-1:k).
88 If UPLO = 'L', then A = L*D*L', where
90 L = P(1)*L(1)* ... *P(k)*L(k)* ...,
91 i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n
92 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
93 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
94 by IPIV(k), and L(k) is a unit lower triangular matrix, such that if
95 the diagonal block D(k) is of order s (s = 1 or 2), then
96 ( I 0 0 ) k-1
98 L(k) = ( 0 I 0 ) s
99 ( 0 v I ) n-k-s+1
100 k-1 s n-k-s+1
101 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s =
103 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and
104 A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
105 LAPACK routine (version 3.1) November 2006 DSPTRF(1)