1CHETD2(1) LAPACK routine (version 3.2) CHETD2(1)
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6 CHETD2 - reduces a complex Hermitian matrix A to real symmetric tridi‐
7 agonal 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.
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25 UPLO (input) CHARACTER*1
26 Specifies whether the upper or lower triangular part of the
27 Hermitian matrix A is stored:
28 = 'U': Upper triangular
29 = 'L': Lower triangular
30 N (input) INTEGER
32 The order of the matrix A. N >= 0.
33 A (input/output) COMPLEX array, dimension (LDA,N)
35 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
36 n-by-n upper triangular part of A contains the upper triangular
37 part of the matrix A, and the strictly lower triangular part of
38 A is not referenced. If UPLO = 'L', the leading n-by-n lower
39 triangular part of A contains the lower triangular part of the
40 matrix A, and the strictly upper triangular part of A is not
41 referenced. On exit, if UPLO = 'U', the diagonal and first
42 superdiagonal of A are overwritten by the corresponding ele‐
43 ments of the tridiagonal matrix T, and the elements above the
44 first superdiagonal, with the array TAU, represent the unitary
45 matrix Q as a product of elementary reflectors; if UPLO = 'L',
46 the diagonal and first subdiagonal of A are over- written by
47 the corresponding elements of the tridiagonal matrix T, and the
48 elements below the first subdiagonal, with the array TAU, rep‐
49 resent the unitary matrix Q as a product of elementary reflec‐
50 tors. See Further Details. LDA (input) INTEGER The leading
51 dimension of the array A. LDA >= max(1,N).
52 D (output) REAL array, dimension (N)
54 The diagonal elements of the tridiagonal matrix T: D(i) =
55 A(i,i).
56 E (output) REAL array, dimension (N-1)
58 The off-diagonal elements of the tridiagonal matrix T: E(i) =
59 A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
60 TAU (output) COMPLEX array, dimension (N-1)
62 The scalar factors of the elementary reflectors (see Further
63 Details).
64 INFO (output) INTEGER
66 = 0: successful exit
67 < 0: if INFO = -i, the i-th argument had an illegal value.
68
70 If UPLO = 'U', the matrix Q is represented as a product of elementary
71 reflectors
72 Q = H(n-1) . . . H(2) H(1).
73 Each H(i) has the form
74 H(i) = I - tau * v * v'
75 where tau is a complex scalar, and v is a complex vector with v(i+1:n)
76 = 0 and v(i) = 1; v(1:i-1) is stored on exit in
77 A(1:i-1,i+1), and tau in TAU(i).
78 If UPLO = 'L', the matrix Q is represented as a product of elementary
79 reflectors
80 Q = H(1) H(2) . . . H(n-1).
81 Each H(i) has the form
82 H(i) = I - tau * v * v'
83 where tau is a complex scalar, and v is a complex vector with v(1:i) =
84 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in
85 TAU(i).
86 The contents of A on exit are illustrated by the following examples
87 with n = 5:
88 if UPLO = 'U': if UPLO = 'L':
89 ( d e v2 v3 v4 ) ( d )
90 ( d e v3 v4 ) ( e d )
91 ( d e v4 ) ( v1 e d )
92 ( d e ) ( v1 v2 e d )
93 ( d ) ( v1 v2 v3 e d ) where d
94 and e denote diagonal and off-diagonal elements of T, and vi denotes an
95 element of the vector defining H(i).
96 LAPACK routine (version 3.2) November 2008 CHETD2(1)