1DSYTRD(1) LAPACK routine (version 3.1) DSYTRD(1)
2
6 DSYTRD - a real symmetric matrix A to real symmetric tridiagonal form T
7 by an orthogonal similarity transformation
8
10 SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, LDA, LWORK, N
15 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
17 WORK( * )
18
20 DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal
21 form T by an orthogonal similarity transformation: Q**T * A * Q = T.
22
25 UPLO (input) CHARACTER*1
26 = 'U': Upper triangle of A is stored;
27 = 'L': Lower triangle of A is stored.
28 N (input) INTEGER
30 The order of the matrix A. N >= 0.
31 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
33 On entry, the symmetric matrix A. If UPLO = 'U', the leading
34 N-by-N upper triangular part of A contains the upper triangular
35 part of the matrix A, and the strictly lower triangular part of
36 A is not referenced. If UPLO = 'L', the leading N-by-N lower
37 triangular part of A contains the lower triangular part of the
38 matrix A, and the strictly upper triangular part of A is not
39 referenced. On exit, if UPLO = 'U', the diagonal and first
40 superdiagonal of A are overwritten by the corresponding ele‐
41 ments of the tridiagonal matrix T, and the elements above the
42 first superdiagonal, with the array TAU, represent the orthogo‐
43 nal matrix Q as a product of elementary reflectors; if UPLO =
44 'L', the diagonal and first subdiagonal of A are over- written
45 by the corresponding elements of the tridiagonal matrix T, and
46 the elements below the first subdiagonal, with the array TAU,
47 represent the orthogonal matrix Q as a product of elementary
48 reflectors. See Further Details. LDA (input) INTEGER The
49 leading dimension of the array A. LDA >= max(1,N).
50 D (output) DOUBLE PRECISION array, dimension (N)
52 The diagonal elements of the tridiagonal matrix T: D(i) =
53 A(i,i).
54 E (output) DOUBLE PRECISION array, dimension (N-1)
56 The off-diagonal elements of the tridiagonal matrix T: E(i) =
57 A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
58 TAU (output) DOUBLE PRECISION array, dimension (N-1)
60 The scalar factors of the elementary reflectors (see Further
61 Details).
62 WORK (workspace/output) DOUBLE PRECISION array, dimension
64 (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
79
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 real scalar, and v is a real vector with
91 v(i+1:n) = 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 real scalar, and v is a real vector with
104 v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
105 and tau in 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 DSYTRD(1)