1CHETD2(1) LAPACK routine (version 3.1) CHETD2(1)
2
6 CHETD2 - a complex Hermitian matrix A to real symmetric tridiagonal
7 form T by a unitary similarity transformation
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10 SUBROUTINE CHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, LDA, N
15 REAL D( * ), E( * )
17 COMPLEX A( LDA, * ), TAU( * )
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21 CHETD2 reduces a complex Hermitian matrix A to real symmetric tridiago‐
22 nal form T by a unitary similarity transformation: Q' * A * Q = T.
23
26 UPLO (input) CHARACTER*1
27 Specifies whether the upper or lower triangular part of the
28 Hermitian matrix A is stored:
29 = 'U': Upper triangular
30 = 'L': Lower triangular
31 N (input) INTEGER
33 The order of the matrix A. N >= 0.
34 A (input/output) COMPLEX array, dimension (LDA,N)
36 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
37 n-by-n upper triangular part of A contains the upper triangular
38 part of the matrix A, and the strictly lower triangular part of
39 A is not referenced. If UPLO = 'L', the leading n-by-n lower
40 triangular part of A contains the lower triangular part of the
41 matrix A, and the strictly upper triangular part of A is not
42 referenced. On exit, if UPLO = 'U', the diagonal and first
43 superdiagonal of A are overwritten by the corresponding ele‐
44 ments of the tridiagonal matrix T, and the elements above the
45 first superdiagonal, with the array TAU, represent the unitary
46 matrix Q as a product of elementary reflectors; if UPLO = 'L',
47 the diagonal and first subdiagonal of A are over- written by
48 the corresponding elements of the tridiagonal matrix T, and the
49 elements below the first subdiagonal, with the array TAU, rep‐
50 resent the unitary matrix Q as a product of elementary reflec‐
51 tors. See Further Details. LDA (input) INTEGER The leading
52 dimension of the array A. LDA >= max(1,N).
53 D (output) REAL array, dimension (N)
55 The diagonal elements of the tridiagonal matrix T: D(i) =
56 A(i,i).
57 E (output) REAL array, dimension (N-1)
59 The off-diagonal elements of the tridiagonal matrix T: E(i) =
60 A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
61 TAU (output) COMPLEX array, dimension (N-1)
63 The scalar factors of the elementary reflectors (see Further
64 Details).
65 INFO (output) INTEGER
67 = 0: successful exit
68 < 0: if INFO = -i, the i-th argument had an illegal value.
69
71 If UPLO = 'U', the matrix Q is represented as a product of elementary
72 reflectors
73 Q = H(n-1) . . . H(2) H(1).
75 Each H(i) has the form
77 H(i) = I - tau * v * v'
79 where tau is a complex scalar, and v is a complex vector with v(i+1:n)
81 = 0 and v(i) = 1; v(1:i-1) is stored on exit in
82 A(1:i-1,i+1), and tau in TAU(i).
83 If UPLO = 'L', the matrix Q is represented as a product of elementary
85 reflectors
86 Q = H(1) H(2) . . . H(n-1).
88 Each H(i) has the form
90 H(i) = I - tau * v * v'
92 where tau is a complex scalar, and v is a complex vector with v(1:i) =
94 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in
95 TAU(i).
96 The contents of A on exit are illustrated by the following examples
98 with n = 5:
99 if UPLO = 'U': if UPLO = 'L':
101 ( d e v2 v3 v4 ) ( d )
103 ( d e v3 v4 ) ( e d )
104 ( d e v4 ) ( v1 e d )
105 ( d e ) ( v1 v2 e d )
106 ( d ) ( v1 v2 v3 e d )
107 where d and e denote diagonal and off-diagonal elements of T, and vi
109 denotes an element of the vector defining H(i).
110 LAPACK routine (version 3.1) November 2006 CHETD2(1)