1ZHPTRD(1) LAPACK routine (version 3.2) ZHPTRD(1)
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6 ZHPTRD - reduces a complex Hermitian matrix A stored in packed form to
7 real symmetric tridiagonal form T by a unitary similarity transforma‐
8 tion
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11 SUBROUTINE ZHPTRD( UPLO, N, AP, D, E, TAU, INFO )
12 CHARACTER UPLO
14 INTEGER INFO, N
16 DOUBLE PRECISION D( * ), E( * )
18 COMPLEX*16 AP( * ), TAU( * )
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22 ZHPTRD reduces a complex Hermitian matrix A stored in packed form to
23 real symmetric tridiagonal form T by a unitary similarity transforma‐
24 tion: Q**H * A * Q = T.
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27 UPLO (input) CHARACTER*1
28 = 'U': Upper triangle of A is stored;
29 = 'L': Lower triangle of A is stored.
30 N (input) INTEGER
32 The order of the matrix A. N >= 0.
33 AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
35 On entry, the upper or lower triangle of the Hermitian matrix
36 A, packed columnwise in a linear array. The j-th column of A
37 is stored in the array AP as follows: if UPLO = 'U', AP(i +
38 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
39 (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, if UPLO = 'U',
40 the diagonal and first superdiagonal of A are overwritten by
41 the corresponding elements of the tridiagonal matrix T, and the
42 elements above the first superdiagonal, with the array TAU,
43 represent the unitary matrix Q as a product of elementary
44 reflectors; if UPLO = 'L', the diagonal and first subdiagonal
45 of A are over- written by the corresponding elements of the
46 tridiagonal matrix T, and the elements below the first subdiag‐
47 onal, with the array TAU, represent the unitary matrix Q as a
48 product of elementary reflectors. See Further Details. D
49 (output) DOUBLE PRECISION array, dimension (N) The diagonal
50 elements of the tridiagonal matrix T: D(i) = A(i,i).
51 E (output) DOUBLE PRECISION array, dimension (N-1)
53 The off-diagonal elements of the tridiagonal matrix T: E(i) =
54 A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
55 TAU (output) COMPLEX*16 array, dimension (N-1)
57 The scalar factors of the elementary reflectors (see Further
58 Details).
59 INFO (output) INTEGER
61 = 0: successful exit
62 < 0: if INFO = -i, the i-th argument had an illegal value
63
65 If UPLO = 'U', the matrix Q is represented as a product of elementary
66 reflectors
67 Q = H(n-1) . . . H(2) H(1).
68 Each H(i) has the form
69 H(i) = I - tau * v * v'
70 where tau is a complex scalar, and v is a complex vector with v(i+1:n)
71 = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP, overwriting
72 A(1:i-1,i+1), and tau is stored in TAU(i).
73 If UPLO = 'L', the matrix Q is represented as a product of elementary
74 reflectors
75 Q = H(1) H(2) . . . H(n-1).
76 Each H(i) has the form
77 H(i) = I - tau * v * v'
78 where tau is a complex scalar, and v is a complex vector with v(1:i) =
79 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, overwriting
80 A(i+2:n,i), and tau is stored in TAU(i).
81 LAPACK routine (version 3.2) November 2008 ZHPTRD(1)