1CLATRD(1) LAPACK auxiliary routine (version 3.1) CLATRD(1)
2
6 CLATRD - NB rows and columns of a complex Hermitian matrix A to Hermit‐
7 ian tridiagonal form by a unitary similarity transformation Q' * A * Q,
8 and returns the matrices V and W which are needed to apply the trans‐
9 formation to the unreduced part of A
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12 SUBROUTINE CLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
13 CHARACTER UPLO
15 INTEGER LDA, LDW, N, NB
17 REAL E( * )
19 COMPLEX A( LDA, * ), TAU( * ), W( LDW, * )
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23 CLATRD reduces NB rows and columns of a complex Hermitian matrix A to
24 Hermitian tridiagonal form by a unitary similarity transformation Q' *
25 A * Q, and returns the matrices V and W which are needed to apply the
26 transformation to the unreduced part of A.
27 If UPLO = 'U', CLATRD reduces the last NB rows and columns of a matrix,
29 of which the upper triangle is supplied;
30 if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
31 matrix, of which the lower triangle is supplied.
32 This is an auxiliary routine called by CHETRD.
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37 UPLO (input) CHARACTER*1
38 Specifies whether the upper or lower triangular part of the
39 Hermitian matrix A is stored:
40 = 'U': Upper triangular
41 = 'L': Lower triangular
42 N (input) INTEGER
44 The order of the matrix A.
45 NB (input) INTEGER
47 The number of rows and columns to be reduced.
48 A (input/output) COMPLEX array, dimension (LDA,N)
50 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
51 n-by-n upper triangular part of A contains the upper triangular
52 part of the matrix A, and the strictly lower triangular part of
53 A is not referenced. If UPLO = 'L', the leading n-by-n lower
54 triangular part of A contains the lower triangular part of the
55 matrix A, and the strictly upper triangular part of A is not
56 referenced. On exit: if UPLO = 'U', the last NB columns have
57 been reduced to tridiagonal form, with the diagonal elements
58 overwriting the diagonal elements of A; the elements above the
59 diagonal with the array TAU, represent the unitary matrix Q as
60 a product of elementary reflectors; if UPLO = 'L', the first NB
61 columns have been reduced to tridiagonal form, with the diago‐
62 nal elements overwriting the diagonal elements of A; the ele‐
63 ments below the diagonal with the array TAU, represent the
64 unitary matrix Q as a product of elementary reflectors. See
65 Further Details. LDA (input) INTEGER The leading dimension
66 of the array A. LDA >= max(1,N).
67 E (output) REAL array, dimension (N-1)
69 If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements
70 of the last NB columns of the reduced matrix; if UPLO = 'L',
71 E(1:nb) contains the subdiagonal elements of the first NB col‐
72 umns of the reduced matrix.
73 TAU (output) COMPLEX array, dimension (N-1)
75 The scalar factors of the elementary reflectors, stored in
76 TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
77 See Further Details. W (output) COMPLEX array, dimension
78 (LDW,NB) The n-by-nb matrix W required to update the unreduced
79 part of A.
80 LDW (input) INTEGER
82 The leading dimension of the array W. LDW >= max(1,N).
83
85 If UPLO = 'U', the matrix Q is represented as a product of elementary
86 reflectors
87 Q = H(n) H(n-1) . . . H(n-nb+1).
89 Each H(i) has the form
91 H(i) = I - tau * v * v'
93 where tau is a complex scalar, and v is a complex vector with v(i:n) =
95 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in
96 TAU(i-1).
97 If UPLO = 'L', the matrix Q is represented as a product of elementary
99 reflectors
100 Q = H(1) H(2) . . . H(nb).
102 Each H(i) has the form
104 H(i) = I - tau * v * v'
106 where tau is a complex scalar, and v is a complex vector with v(1:i) =
108 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in
109 TAU(i).
110 The elements of the vectors v together form the n-by-nb matrix V which
112 is needed, with W, to apply the transformation to the unreduced part of
113 the matrix, using a Hermitian rank-2k update of the form: A := A - V*W'
114 - W*V'.
115 The contents of A on exit are illustrated by the following examples
117 with n = 5 and nb = 2:
118 if UPLO = 'U': if UPLO = 'L':
120 ( a a a v4 v5 ) ( d )
122 ( a a v4 v5 ) ( 1 d )
123 ( a 1 v5 ) ( v1 1 a )
124 ( d 1 ) ( v1 v2 a a )
125 ( d ) ( v1 v2 a a a )
126 where d denotes a diagonal element of the reduced matrix, a denotes an
128 element of the original matrix that is unchanged, and vi denotes an
129 element of the vector defining H(i).
130 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 CLATRD(1)