1DLATRD(1) LAPACK auxiliary routine (version 3.1) DLATRD(1)
2
6 DLATRD - NB rows and columns of a real symmetric matrix A to symmetric
7 tridiagonal form by an orthogonal 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 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.
25 If UPLO = 'U', DLATRD reduces the last NB rows and columns of a matrix,
27 of which the upper triangle is supplied;
28 if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
29 matrix, of which the lower triangle is supplied.
30 This is an auxiliary routine called by DSYTRD.
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35 UPLO (input) CHARACTER*1
36 Specifies whether the upper or lower triangular part of the
37 symmetric matrix A is stored:
38 = 'U': Upper triangular
39 = 'L': Lower triangular
40 N (input) INTEGER
42 The order of the matrix A.
43 NB (input) INTEGER
45 The number of rows and columns to be reduced.
46 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
48 On entry, the symmetric matrix A. If UPLO = 'U', the leading
49 n-by-n upper triangular part of A contains the upper triangular
50 part of the matrix A, and the strictly lower triangular part of
51 A is not referenced. If UPLO = 'L', the leading n-by-n lower
52 triangular part of A contains the lower triangular part of the
53 matrix A, and the strictly upper triangular part of A is not
54 referenced. On exit: if UPLO = 'U', the last NB columns have
55 been reduced to tridiagonal form, with the diagonal elements
56 overwriting the diagonal elements of A; the elements above the
57 diagonal with the array TAU, represent the orthogonal matrix Q
58 as a product of elementary reflectors; if UPLO = 'L', the first
59 NB columns have been reduced to tridiagonal form, with the
60 diagonal elements overwriting the diagonal elements of A; the
61 elements below the diagonal with the array TAU, represent the
62 orthogonal matrix Q as a product of elementary reflectors. See
63 Further Details. LDA (input) INTEGER The leading dimension
64 of the array A. LDA >= (1,N).
65 E (output) DOUBLE PRECISION array, dimension (N-1)
67 If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements
68 of the last NB columns of the reduced matrix; if UPLO = 'L',
69 E(1:nb) contains the subdiagonal elements of the first NB col‐
70 umns of the reduced matrix.
71 TAU (output) DOUBLE PRECISION array, dimension (N-1)
73 The scalar factors of the elementary reflectors, stored in
74 TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
75 See Further Details. W (output) DOUBLE PRECISION array,
76 dimension (LDW,NB) The n-by-nb matrix W required to update the
77 unreduced part of A.
78 LDW (input) INTEGER
80 The leading dimension of the array W. LDW >= max(1,N).
81
83 If UPLO = 'U', the matrix Q is represented as a product of elementary
84 reflectors
85 Q = H(n) H(n-1) . . . H(n-nb+1).
87 Each H(i) has the form
89 H(i) = I - tau * v * v'
91 where tau is a real scalar, and v is a real vector with
93 v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
94 and tau in TAU(i-1).
95 If UPLO = 'L', the matrix Q is represented as a product of elementary
97 reflectors
98 Q = H(1) H(2) . . . H(nb).
100 Each H(i) has the form
102 H(i) = I - tau * v * v'
104 where tau is a real scalar, and v is a real vector with
106 v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
107 and tau in TAU(i).
108 The elements of the vectors v together form the n-by-nb matrix V which
110 is needed, with W, to apply the transformation to the unreduced part of
111 the matrix, using a symmetric rank-2k update of the form: A := A - V*W'
112 - W*V'.
113 The contents of A on exit are illustrated by the following examples
115 with n = 5 and nb = 2:
116 if UPLO = 'U': if UPLO = 'L':
118 ( a a a v4 v5 ) ( d )
120 ( a a v4 v5 ) ( 1 d )
121 ( a 1 v5 ) ( v1 1 a )
122 ( d 1 ) ( v1 v2 a a )
123 ( d ) ( v1 v2 a a a )
124 where d denotes a diagonal element of the reduced matrix, a denotes an
126 element of the original matrix that is unchanged, and vi denotes an
127 element of the vector defining H(i).
128 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 DLATRD(1)