1DLATRD(1) LAPACK auxiliary routine (version 3.2) DLATRD(1)
2
6 DLATRD - reduces NB rows and columns of a real symmetric matrix A to
7 symmetric tridiagonal form by an orthogonal similarity transformation
8 Q' * A * Q, and returns the matrices V and W which are needed to apply
9 the transformation to the unreduced part of A
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12 SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
13 CHARACTER UPLO
15 INTEGER LDA, LDW, N, NB
17 DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
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21 DLATRD reduces NB rows and columns of a real symmetric matrix A to sym‐
22 metric tridiagonal form by an orthogonal similarity transformation Q' *
23 A * Q, and returns the matrices V and W which are needed to apply the
24 transformation to the unreduced part of A. If UPLO = 'U', DLATRD
25 reduces the last NB rows and columns of a matrix, of which the upper
26 triangle is supplied;
27 if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
28 matrix, of which the lower triangle is supplied.
29 This is an auxiliary routine called by DSYTRD.
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32 UPLO (input) CHARACTER*1
33 Specifies whether the upper or lower triangular part of the
34 symmetric matrix A is stored:
35 = 'U': Upper triangular
36 = 'L': Lower triangular
37 N (input) INTEGER
39 The order of the matrix A.
40 NB (input) INTEGER
42 The number of rows and columns to be reduced.
43 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
45 On entry, the symmetric matrix A. If UPLO = 'U', the leading
46 n-by-n upper triangular part of A contains the upper triangular
47 part of the matrix A, and the strictly lower triangular part of
48 A is not referenced. If UPLO = 'L', the leading n-by-n lower
49 triangular part of A contains the lower triangular part of the
50 matrix A, and the strictly upper triangular part of A is not
51 referenced. On exit: if UPLO = 'U', the last NB columns have
52 been reduced to tridiagonal form, with the diagonal elements
53 overwriting the diagonal elements of A; the elements above the
54 diagonal with the array TAU, represent the orthogonal matrix Q
55 as a product of elementary reflectors; if UPLO = 'L', the first
56 NB columns have been reduced to tridiagonal form, with the
57 diagonal elements overwriting the diagonal elements of A; the
58 elements below the diagonal with the array TAU, represent the
59 orthogonal matrix Q as a product of elementary reflectors. See
60 Further Details. LDA (input) INTEGER The leading dimension
61 of the array A. LDA >= (1,N).
62 E (output) DOUBLE PRECISION array, dimension (N-1)
64 If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements
65 of the last NB columns of the reduced matrix; if UPLO = 'L',
66 E(1:nb) contains the subdiagonal elements of the first NB col‐
67 umns of the reduced matrix.
68 TAU (output) DOUBLE PRECISION array, dimension (N-1)
70 The scalar factors of the elementary reflectors, stored in
71 TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
72 See Further Details. W (output) DOUBLE PRECISION array,
73 dimension (LDW,NB) The n-by-nb matrix W required to update the
74 unreduced part of A.
75 LDW (input) INTEGER
77 The leading dimension of the array W. LDW >= max(1,N).
78
80 If UPLO = 'U', the matrix Q is represented as a product of elementary
81 reflectors
82 Q = H(n) H(n-1) . . . H(n-nb+1).
83 Each H(i) has the form
84 H(i) = I - tau * v * v'
85 where tau is a real scalar, and v is a real vector with
86 v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
87 and tau in TAU(i-1).
88 If UPLO = 'L', the matrix Q is represented as a product of elementary
89 reflectors
90 Q = H(1) H(2) . . . H(nb).
91 Each H(i) has the form
92 H(i) = I - tau * v * v'
93 where tau is a real scalar, and v is a real vector with
94 v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
95 and tau in TAU(i).
96 The elements of the vectors v together form the n-by-nb matrix V which
97 is needed, with W, to apply the transformation to the unreduced part of
98 the matrix, using a symmetric rank-2k update of the form: A := A - V*W'
99 - W*V'.
100 The contents of A on exit are illustrated by the following examples
101 with n = 5 and nb = 2:
102 if UPLO = 'U': if UPLO = 'L':
103 ( a a a v4 v5 ) ( d )
104 ( a a v4 v5 ) ( 1 d )
105 ( a 1 v5 ) ( v1 1 a )
106 ( d 1 ) ( v1 v2 a a )
107 ( d ) ( v1 v2 a a a ) where d
108 denotes a diagonal element of the reduced matrix, a denotes an element
109 of the original matrix that is unchanged, and vi denotes an element of
110 the vector defining H(i).
111 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 DLATRD(1)