1: SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
2: *
3: * -- LAPACK auxiliary routine (version 3.2) --
4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6: * November 2006
7: *
8: * .. Scalar Arguments ..
9: CHARACTER UPLO
10: INTEGER LDA, LDW, N, NB
11: * ..
12: * .. Array Arguments ..
13: DOUBLE PRECISION E( * )
14: COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * )
15: * ..
16: *
17: * Purpose
18: * =======
19: *
20: * ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
21: * Hermitian tridiagonal form by a unitary similarity
22: * transformation Q' * A * Q, and returns the matrices V and W which are
23: * needed to apply the transformation to the unreduced part of A.
24: *
25: * If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a
26: * matrix, of which the upper triangle is supplied;
27: * if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
28: * matrix, of which the lower triangle is supplied.
29: *
30: * This is an auxiliary routine called by ZHETRD.
31: *
32: * Arguments
33: * =========
34: *
35: * UPLO (input) CHARACTER*1
36: * Specifies whether the upper or lower triangular part of the
37: * Hermitian matrix A is stored:
38: * = 'U': Upper triangular
39: * = 'L': Lower triangular
40: *
41: * N (input) INTEGER
42: * The order of the matrix A.
43: *
44: * NB (input) INTEGER
45: * The number of rows and columns to be reduced.
46: *
47: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
48: * On entry, the Hermitian matrix A. If UPLO = 'U', the leading
49: * n-by-n upper triangular part of A contains the upper
50: * triangular part of the matrix A, and the strictly lower
51: * triangular part of A is not referenced. If UPLO = 'L', the
52: * leading n-by-n lower triangular part of A contains the lower
53: * triangular part of the matrix A, and the strictly upper
54: * triangular part of A is not referenced.
55: * On exit:
56: * if UPLO = 'U', the last NB columns have been reduced to
57: * tridiagonal form, with the diagonal elements overwriting
58: * the diagonal elements of A; the elements above the diagonal
59: * with the array TAU, represent the unitary matrix Q as a
60: * product of elementary reflectors;
61: * if UPLO = 'L', the first NB columns have been reduced to
62: * tridiagonal form, with the diagonal elements overwriting
63: * the diagonal elements of A; the elements below the diagonal
64: * with the array TAU, represent the unitary matrix Q as a
65: * product of elementary reflectors.
66: * See Further Details.
67: *
68: * LDA (input) INTEGER
69: * The leading dimension of the array A. LDA >= max(1,N).
70: *
71: * E (output) DOUBLE PRECISION array, dimension (N-1)
72: * If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
73: * elements of the last NB columns of the reduced matrix;
74: * if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
75: * the first NB columns of the reduced matrix.
76: *
77: * TAU (output) COMPLEX*16 array, dimension (N-1)
78: * The scalar factors of the elementary reflectors, stored in
79: * TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
80: * See Further Details.
81: *
82: * W (output) COMPLEX*16 array, dimension (LDW,NB)
83: * The n-by-nb matrix W required to update the unreduced part
84: * of A.
85: *
86: * LDW (input) INTEGER
87: * The leading dimension of the array W. LDW >= max(1,N).
88: *
89: * Further Details
90: * ===============
91: *
92: * If UPLO = 'U', the matrix Q is represented as a product of elementary
93: * reflectors
94: *
95: * Q = H(n) H(n-1) . . . H(n-nb+1).
96: *
97: * Each H(i) has the form
98: *
99: * H(i) = I - tau * v * v'
100: *
101: * where tau is a complex scalar, and v is a complex vector with
102: * v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
103: * and tau in TAU(i-1).
104: *
105: * If UPLO = 'L', the matrix Q is represented as a product of elementary
106: * reflectors
107: *
108: * Q = H(1) H(2) . . . H(nb).
109: *
110: * Each H(i) has the form
111: *
112: * H(i) = I - tau * v * v'
113: *
114: * where tau is a complex scalar, and v is a complex vector with
115: * v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
116: * and tau in TAU(i).
117: *
118: * The elements of the vectors v together form the n-by-nb matrix V
119: * which is needed, with W, to apply the transformation to the unreduced
120: * part of the matrix, using a Hermitian rank-2k update of the form:
121: * A := A - V*W' - W*V'.
122: *
123: * The contents of A on exit are illustrated by the following examples
124: * with n = 5 and nb = 2:
125: *
126: * if UPLO = 'U': if UPLO = 'L':
127: *
128: * ( a a a v4 v5 ) ( d )
129: * ( a a v4 v5 ) ( 1 d )
130: * ( a 1 v5 ) ( v1 1 a )
131: * ( d 1 ) ( v1 v2 a a )
132: * ( d ) ( v1 v2 a a a )
133: *
134: * where d denotes a diagonal element of the reduced matrix, a denotes
135: * an element of the original matrix that is unchanged, and vi denotes
136: * an element of the vector defining H(i).
137: *
138: * =====================================================================
139: *
140: * .. Parameters ..
141: COMPLEX*16 ZERO, ONE, HALF
142: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
143: $ ONE = ( 1.0D+0, 0.0D+0 ),
144: $ HALF = ( 0.5D+0, 0.0D+0 ) )
145: * ..
146: * .. Local Scalars ..
147: INTEGER I, IW
148: COMPLEX*16 ALPHA
149: * ..
150: * .. External Subroutines ..
151: EXTERNAL ZAXPY, ZGEMV, ZHEMV, ZLACGV, ZLARFG, ZSCAL
152: * ..
153: * .. External Functions ..
154: LOGICAL LSAME
155: COMPLEX*16 ZDOTC
156: EXTERNAL LSAME, ZDOTC
157: * ..
158: * .. Intrinsic Functions ..
159: INTRINSIC DBLE, MIN
160: * ..
161: * .. Executable Statements ..
162: *
163: * Quick return if possible
164: *
165: IF( N.LE.0 )
166: $ RETURN
167: *
168: IF( LSAME( UPLO, 'U' ) ) THEN
169: *
170: * Reduce last NB columns of upper triangle
171: *
172: DO 10 I = N, N - NB + 1, -1
173: IW = I - N + NB
174: IF( I.LT.N ) THEN
175: *
176: * Update A(1:i,i)
177: *
178: A( I, I ) = DBLE( A( I, I ) )
179: CALL ZLACGV( N-I, W( I, IW+1 ), LDW )
180: CALL ZGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
181: $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
182: CALL ZLACGV( N-I, W( I, IW+1 ), LDW )
183: CALL ZLACGV( N-I, A( I, I+1 ), LDA )
184: CALL ZGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
185: $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
186: CALL ZLACGV( N-I, A( I, I+1 ), LDA )
187: A( I, I ) = DBLE( A( I, I ) )
188: END IF
189: IF( I.GT.1 ) THEN
190: *
191: * Generate elementary reflector H(i) to annihilate
192: * A(1:i-2,i)
193: *
194: ALPHA = A( I-1, I )
195: CALL ZLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) )
196: E( I-1 ) = ALPHA
197: A( I-1, I ) = ONE
198: *
199: * Compute W(1:i-1,i)
200: *
201: CALL ZHEMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
202: $ ZERO, W( 1, IW ), 1 )
203: IF( I.LT.N ) THEN
204: CALL ZGEMV( 'Conjugate transpose', I-1, N-I, ONE,
205: $ W( 1, IW+1 ), LDW, A( 1, I ), 1, ZERO,
206: $ W( I+1, IW ), 1 )
207: CALL ZGEMV( 'No transpose', I-1, N-I, -ONE,
208: $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
209: $ W( 1, IW ), 1 )
210: CALL ZGEMV( 'Conjugate transpose', I-1, N-I, ONE,
211: $ A( 1, I+1 ), LDA, A( 1, I ), 1, ZERO,
212: $ W( I+1, IW ), 1 )
213: CALL ZGEMV( 'No transpose', I-1, N-I, -ONE,
214: $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
215: $ W( 1, IW ), 1 )
216: END IF
217: CALL ZSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
218: ALPHA = -HALF*TAU( I-1 )*ZDOTC( I-1, W( 1, IW ), 1,
219: $ A( 1, I ), 1 )
220: CALL ZAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
221: END IF
222: *
223: 10 CONTINUE
224: ELSE
225: *
226: * Reduce first NB columns of lower triangle
227: *
228: DO 20 I = 1, NB
229: *
230: * Update A(i:n,i)
231: *
232: A( I, I ) = DBLE( A( I, I ) )
233: CALL ZLACGV( I-1, W( I, 1 ), LDW )
234: CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
235: $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
236: CALL ZLACGV( I-1, W( I, 1 ), LDW )
237: CALL ZLACGV( I-1, A( I, 1 ), LDA )
238: CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
239: $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
240: CALL ZLACGV( I-1, A( I, 1 ), LDA )
241: A( I, I ) = DBLE( A( I, I ) )
242: IF( I.LT.N ) THEN
243: *
244: * Generate elementary reflector H(i) to annihilate
245: * A(i+2:n,i)
246: *
247: ALPHA = A( I+1, I )
248: CALL ZLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1,
249: $ TAU( I ) )
250: E( I ) = ALPHA
251: A( I+1, I ) = ONE
252: *
253: * Compute W(i+1:n,i)
254: *
255: CALL ZHEMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
256: $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
257: CALL ZGEMV( 'Conjugate transpose', N-I, I-1, ONE,
258: $ W( I+1, 1 ), LDW, A( I+1, I ), 1, ZERO,
259: $ W( 1, I ), 1 )
260: CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),
261: $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
262: CALL ZGEMV( 'Conjugate transpose', N-I, I-1, ONE,
263: $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO,
264: $ W( 1, I ), 1 )
265: CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),
266: $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
267: CALL ZSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
268: ALPHA = -HALF*TAU( I )*ZDOTC( N-I, W( I+1, I ), 1,
269: $ A( I+1, I ), 1 )
270: CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )
271: END IF
272: *
273: 20 CONTINUE
274: END IF
275: *
276: RETURN
277: *
278: * End of ZLATRD
279: *
280: END
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