1DSYTRD(1) LAPACK routine (version 3.2) DSYTRD(1)
2
6 DSYTRD - reduces a real symmetric matrix A to real symmetric tridiago‐
7 nal form T by an orthogonal similarity transformation
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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
24 UPLO (input) CHARACTER*1
25 = 'U': Upper triangle of A is stored;
26 = 'L': Lower triangle of A is stored.
27 N (input) INTEGER
29 The order of the matrix A. N >= 0.
30 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
32 On entry, the symmetric matrix A. If UPLO = 'U', the leading
33 N-by-N upper triangular part of A contains the upper triangular
34 part of the matrix A, and the strictly lower triangular part of
35 A is not referenced. If UPLO = 'L', the leading N-by-N lower
36 triangular part of A contains the lower triangular part of the
37 matrix A, and the strictly upper triangular part of A is not
38 referenced. On exit, if UPLO = 'U', the diagonal and first
39 superdiagonal of A are overwritten by the corresponding ele‐
40 ments of the tridiagonal matrix T, and the elements above the
41 first superdiagonal, with the array TAU, represent the orthogo‐
42 nal matrix Q as a product of elementary reflectors; if UPLO =
43 'L', the diagonal and first subdiagonal of A are over- written
44 by the corresponding elements of the tridiagonal matrix T, and
45 the elements below the first subdiagonal, with the array TAU,
46 represent the orthogonal matrix Q as a product of elementary
47 reflectors. See Further Details. LDA (input) INTEGER The
48 leading dimension of the array A. LDA >= max(1,N).
49 D (output) DOUBLE PRECISION array, dimension (N)
51 The diagonal elements of the tridiagonal matrix T: D(i) =
52 A(i,i).
53 E (output) DOUBLE PRECISION array, dimension (N-1)
55 The off-diagonal elements of the tridiagonal matrix T: E(i) =
56 A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
57 TAU (output) DOUBLE PRECISION array, dimension (N-1)
59 The scalar factors of the elementary reflectors (see Further
60 Details).
61 WORK (workspace/output) DOUBLE PRECISION array, dimension
63 (MAX(1,LWORK))
64 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
65 LWORK (input) INTEGER
67 The dimension of the array WORK. LWORK >= 1. For optimum per‐
68 formance LWORK >= N*NB, where NB is the optimal blocksize. If
69 LWORK = -1, then a workspace query is assumed; the routine only
70 calculates the optimal size of the WORK array, returns this
71 value as the first entry of the WORK array, and no error mes‐
72 sage related to LWORK is issued by XERBLA.
73 INFO (output) INTEGER
75 = 0: successful exit
76 < 0: if INFO = -i, the i-th argument had an illegal value
77
79 If UPLO = 'U', the matrix Q is represented as a product of elementary
80 reflectors
81 Q = H(n-1) . . . H(2) H(1).
82 Each H(i) has the form
83 H(i) = I - tau * v * v'
84 where tau is a real scalar, and v is a real vector with
85 v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
86 A(1:i-1,i+1), and tau in TAU(i).
87 If UPLO = 'L', the matrix Q is represented as a product of elementary
88 reflectors
89 Q = H(1) H(2) . . . H(n-1).
90 Each H(i) has the form
91 H(i) = I - tau * v * v'
92 where tau is a real scalar, and v is a real vector with
93 v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
94 and tau in TAU(i).
95 The contents of A on exit are illustrated by the following examples
96 with n = 5:
97 if UPLO = 'U': if UPLO = 'L':
98 ( d e v2 v3 v4 ) ( d )
99 ( d e v3 v4 ) ( e d )
100 ( d e v4 ) ( v1 e d )
101 ( d e ) ( v1 v2 e d )
102 ( d ) ( v1 v2 v3 e d ) where d
103 and e denote diagonal and off-diagonal elements of T, and vi denotes an
104 element of the vector defining H(i).
105 LAPACK routine (version 3.2) November 2008 DSYTRD(1)