1ZHETRD(1) LAPACK routine (version 3.1) ZHETRD(1)
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6 ZHETRD - a complex Hermitian matrix A to real symmetric tridiagonal
7 form T by a unitary similarity transformation
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10 SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, LDA, LWORK, N
15 DOUBLE PRECISION D( * ), E( * )
17 COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
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21 ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiago‐
22 nal form T by a unitary similarity transformation: Q**H * A * Q = T.
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26 UPLO (input) CHARACTER*1
27 = 'U': Upper triangle of A is stored;
28 = 'L': Lower triangle of A is stored.
29 N (input) INTEGER
31 The order of the matrix A. N >= 0.
32 A (input/output) COMPLEX*16 array, dimension (LDA,N)
34 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
35 N-by-N upper triangular part of A contains the upper triangular
36 part of the matrix A, and the strictly lower triangular part of
37 A is not referenced. If UPLO = 'L', the leading N-by-N lower
38 triangular part of A contains the lower triangular part of the
39 matrix A, and the strictly upper triangular part of A is not
40 referenced. On exit, if UPLO = 'U', the diagonal and first
41 superdiagonal of A are overwritten by the corresponding ele‐
42 ments of the tridiagonal matrix T, and the elements above the
43 first superdiagonal, with the array TAU, represent the unitary
44 matrix Q as a product of elementary reflectors; if UPLO = 'L',
45 the diagonal and first subdiagonal of A are over- written by
46 the corresponding elements of the tridiagonal matrix T, and the
47 elements below the first subdiagonal, with the array TAU, rep‐
48 resent the unitary matrix Q as a product of elementary reflec‐
49 tors. See Further Details. LDA (input) INTEGER The leading
50 dimension of the array A. LDA >= max(1,N).
51 D (output) DOUBLE PRECISION array, dimension (N)
53 The diagonal elements of the tridiagonal matrix T: D(i) =
54 A(i,i).
55 E (output) DOUBLE PRECISION array, dimension (N-1)
57 The off-diagonal elements of the tridiagonal matrix T: E(i) =
58 A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
59 TAU (output) COMPLEX*16 array, dimension (N-1)
61 The scalar factors of the elementary reflectors (see Further
62 Details).
63 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
65 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
66 LWORK (input) INTEGER
68 The dimension of the array WORK. LWORK >= 1. For optimum per‐
69 formance LWORK >= N*NB, where NB is the optimal blocksize.
70 If LWORK = -1, then a workspace query is assumed; the routine
72 only calculates the optimal size of the WORK array, returns
73 this value as the first entry of the WORK array, and no error
74 message related to LWORK is issued by XERBLA.
75 INFO (output) INTEGER
77 = 0: successful exit
78 < 0: if INFO = -i, the i-th argument had an illegal value
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81 If UPLO = 'U', the matrix Q is represented as a product of elementary
82 reflectors
83 Q = H(n-1) . . . H(2) H(1).
85 Each H(i) has the form
87 H(i) = I - tau * v * v'
89 where tau is a complex scalar, and v is a complex vector with v(i+1:n)
91 = 0 and v(i) = 1; v(1:i-1) is stored on exit in
92 A(1:i-1,i+1), and tau in TAU(i).
93 If UPLO = 'L', the matrix Q is represented as a product of elementary
95 reflectors
96 Q = H(1) H(2) . . . H(n-1).
98 Each H(i) has the form
100 H(i) = I - tau * v * v'
102 where tau is a complex scalar, and v is a complex vector with v(1:i) =
104 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in
105 TAU(i).
106 The contents of A on exit are illustrated by the following examples
108 with n = 5:
109 if UPLO = 'U': if UPLO = 'L':
111 ( d e v2 v3 v4 ) ( d )
113 ( d e v3 v4 ) ( e d )
114 ( d e v4 ) ( v1 e d )
115 ( d e ) ( v1 v2 e d )
116 ( d ) ( v1 v2 v3 e d )
117 where d and e denote diagonal and off-diagonal elements of T, and vi
119 denotes an element of the vector defining H(i).
120 LAPACK routine (version 3.1) November 2006 ZHETRD(1)