1DSYTD2(1) LAPACK routine (version 3.1) DSYTD2(1)
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6 DSYTD2 - a real symmetric matrix A to symmetric tridiagonal form T by
7 an orthogonal similarity transformation
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10 SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, LDA, N
15 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
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19 DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form
20 T by an orthogonal similarity transformation: Q' * A * Q = T.
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24 UPLO (input) CHARACTER*1
25 Specifies whether the upper or lower triangular part of the
26 symmetric matrix A is stored:
27 = 'U': Upper triangular
28 = 'L': Lower triangular
29 N (input) INTEGER
31 The order of the matrix A. N >= 0.
32 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
34 On entry, the symmetric matrix A. If UPLO = 'U', the leading
35 n-by-n upper triangular part of A contains the upper triangular
36 part of the matrix A, and the strictly lower triangular part of
37 A is not referenced. If UPLO = 'L', the leading n-by-n lower
38 triangular part of A contains the lower triangular part of the
39 matrix A, and the strictly upper triangular part of A is not
40 referenced. On exit, if UPLO = 'U', the diagonal and first
41 superdiagonal of A are overwritten by the corresponding ele‐
42 ments of the tridiagonal matrix T, and the elements above the
43 first superdiagonal, with the array TAU, represent the orthogo‐
44 nal matrix Q as a product of elementary reflectors; if UPLO =
45 'L', the diagonal and first subdiagonal of A are over- written
46 by the corresponding elements of the tridiagonal matrix T, and
47 the elements below the first subdiagonal, with the array TAU,
48 represent the orthogonal matrix Q as a product of elementary
49 reflectors. See Further Details. LDA (input) INTEGER The
50 leading dimension of the array A. LDA >= max(1,N).
51 D (output) DOUBLE PRECISION array, dimension (N)
53 The diagonal elements of the tridiagonal matrix T: D(i) =
54 A(i,i).
55 E (output) DOUBLE PRECISION array, dimension (N-1)
57 The off-diagonal elements of the tridiagonal matrix T: E(i) =
58 A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
59 TAU (output) DOUBLE PRECISION array, dimension (N-1)
61 The scalar factors of the elementary reflectors (see Further
62 Details).
63 INFO (output) INTEGER
65 = 0: successful exit
66 < 0: if INFO = -i, the i-th argument had an illegal value.
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69 If UPLO = 'U', the matrix Q is represented as a product of elementary
70 reflectors
71 Q = H(n-1) . . . H(2) H(1).
73 Each H(i) has the form
75 H(i) = I - tau * v * v'
77 where tau is a real scalar, and v is a real vector with
79 v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
80 A(1:i-1,i+1), and tau in TAU(i).
81 If UPLO = 'L', the matrix Q is represented as a product of elementary
83 reflectors
84 Q = H(1) H(2) . . . H(n-1).
86 Each H(i) has the form
88 H(i) = I - tau * v * v'
90 where tau is a real scalar, and v is a real vector with
92 v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
93 and tau in TAU(i).
94 The contents of A on exit are illustrated by the following examples
96 with n = 5:
97 if UPLO = 'U': if UPLO = 'L':
99 ( d e v2 v3 v4 ) ( d )
101 ( d e v3 v4 ) ( e d )
102 ( d e v4 ) ( v1 e d )
103 ( d e ) ( v1 v2 e d )
104 ( d ) ( v1 v2 v3 e d )
105 where d and e denote diagonal and off-diagonal elements of T, and vi
107 denotes an element of the vector defining H(i).
108 LAPACK routine (version 3.1) November 2006 DSYTD2(1)