1SSYTRD(1) LAPACK routine (version 3.1) SSYTRD(1)
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6 SSYTRD - a real symmetric matrix A to real symmetric tridiagonal form T
7 by an orthogonal similarity transformation
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10 SUBROUTINE SSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, LDA, LWORK, N
15 REAL A( LDA, * ), D( * ), E( * ), TAU( * ), WORK( * )
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19 SSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal
20 form T by an orthogonal similarity transformation: Q**T * A * Q = T.
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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) REAL 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) REAL array, dimension (N)
51 The diagonal elements of the tridiagonal matrix T: D(i) =
52 A(i,i).
53 E (output) REAL 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) REAL array, dimension (N-1)
59 The scalar factors of the elementary reflectors (see Further
60 Details).
61 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
63 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
64 LWORK (input) INTEGER
66 The dimension of the array WORK. LWORK >= 1. For optimum per‐
67 formance LWORK >= N*NB, where NB is the optimal blocksize.
68 If LWORK = -1, then a workspace query is assumed; the routine
70 only calculates the optimal size of the WORK array, returns
71 this value as the first entry of the WORK array, and no error
72 message 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).
83 Each H(i) has the form
85 H(i) = I - tau * v * v'
87 where tau is a real scalar, and v is a real vector with
89 v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
90 A(1:i-1,i+1), and tau in TAU(i).
91 If UPLO = 'L', the matrix Q is represented as a product of elementary
93 reflectors
94 Q = H(1) H(2) . . . H(n-1).
96 Each H(i) has the form
98 H(i) = I - tau * v * v'
100 where tau is a real scalar, and v is a real vector with
102 v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
103 and tau in TAU(i).
104 The contents of A on exit are illustrated by the following examples
106 with n = 5:
107 if UPLO = 'U': if UPLO = 'L':
109 ( d e v2 v3 v4 ) ( d )
111 ( d e v3 v4 ) ( e d )
112 ( d e v4 ) ( v1 e d )
113 ( d e ) ( v1 v2 e d )
114 ( d ) ( v1 v2 v3 e d )
115 where d and e denote diagonal and off-diagonal elements of T, and vi
117 denotes an element of the vector defining H(i).
118 LAPACK routine (version 3.1) November 2006 SSYTRD(1)