1CLATRD(1) LAPACK auxiliary routine (version 3.2) CLATRD(1)
2
6 CLATRD - reduces NB rows and columns of a complex Hermitian matrix A to
7 Hermitian tridiagonal form by a unitary similarity transformation Q' *
8 A * Q, and returns the matrices V and W which are needed to apply the
9 transformation 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. If UPLO = 'U', CLATRD
27 reduces the last NB rows and columns of a matrix, of which the upper
28 triangle is supplied;
29 if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
30 matrix, of which the lower triangle is supplied.
31 This is an auxiliary routine called by CHETRD.
32
34 UPLO (input) CHARACTER*1
35 Specifies whether the upper or lower triangular part of the
36 Hermitian matrix A is stored:
37 = 'U': Upper triangular
38 = 'L': Lower triangular
39 N (input) INTEGER
41 The order of the matrix A.
42 NB (input) INTEGER
44 The number of rows and columns to be reduced.
45 A (input/output) COMPLEX array, dimension (LDA,N)
47 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
48 n-by-n upper triangular part of A contains the upper triangular
49 part of the matrix A, and the strictly lower triangular part of
50 A is not referenced. If UPLO = 'L', the leading n-by-n lower
51 triangular part of A contains the lower triangular part of the
52 matrix A, and the strictly upper triangular part of A is not
53 referenced. On exit: if UPLO = 'U', the last NB columns have
54 been reduced to tridiagonal form, with the diagonal elements
55 overwriting the diagonal elements of A; the elements above the
56 diagonal with the array TAU, represent the unitary matrix Q as
57 a product of elementary reflectors; if UPLO = 'L', the first NB
58 columns have been reduced to tridiagonal form, with the diago‐
59 nal elements overwriting the diagonal elements of A; the ele‐
60 ments below the diagonal with the array TAU, represent the
61 unitary matrix Q as a product of elementary reflectors. See
62 Further Details. LDA (input) INTEGER The leading dimension
63 of the array A. LDA >= max(1,N).
64 E (output) REAL array, dimension (N-1)
66 If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements
67 of the last NB columns of the reduced matrix; if UPLO = 'L',
68 E(1:nb) contains the subdiagonal elements of the first NB col‐
69 umns of the reduced matrix.
70 TAU (output) COMPLEX array, dimension (N-1)
72 The scalar factors of the elementary reflectors, stored in
73 TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
74 See Further Details. W (output) COMPLEX array, dimension
75 (LDW,NB) The n-by-nb matrix W required to update the unreduced
76 part of A.
77 LDW (input) INTEGER
79 The leading dimension of the array W. LDW >= max(1,N).
80
82 If UPLO = 'U', the matrix Q is represented as a product of elementary
83 reflectors
84 Q = H(n) H(n-1) . . . H(n-nb+1).
85 Each H(i) has the form
86 H(i) = I - tau * v * v'
87 where tau is a complex scalar, and v is a complex vector with v(i:n) =
88 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in
89 TAU(i-1).
90 If UPLO = 'L', the matrix Q is represented as a product of elementary
91 reflectors
92 Q = H(1) H(2) . . . H(nb).
93 Each H(i) has the form
94 H(i) = I - tau * v * v'
95 where tau is a complex scalar, and v is a complex vector with v(1:i) =
96 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in
97 TAU(i).
98 The elements of the vectors v together form the n-by-nb matrix V which
99 is needed, with W, to apply the transformation to the unreduced part of
100 the matrix, using a Hermitian rank-2k update of the form: A := A - V*W'
101 - W*V'.
102 The contents of A on exit are illustrated by the following examples
103 with n = 5 and nb = 2:
104 if UPLO = 'U': if UPLO = 'L':
105 ( a a a v4 v5 ) ( d )
106 ( a a v4 v5 ) ( 1 d )
107 ( a 1 v5 ) ( v1 1 a )
108 ( d 1 ) ( v1 v2 a a )
109 ( d ) ( v1 v2 a a a ) where d
110 denotes a diagonal element of the reduced matrix, a denotes an element
111 of the original matrix that is unchanged, and vi denotes an element of
112 the vector defining H(i).
113 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 CLATRD(1)