1: *> \brief \b DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLATRD + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrd.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrd.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrd.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER LDA, LDW, N, NB
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DLATRD reduces NB rows and columns of a real symmetric matrix A to
38: *> symmetric tridiagonal form by an orthogonal similarity
39: *> transformation Q**T * A * Q, and returns the matrices V and W which are
40: *> needed to apply the transformation to the unreduced part of A.
41: *>
42: *> If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
43: *> matrix, of which the upper triangle is supplied;
44: *> if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
45: *> matrix, of which the lower triangle is supplied.
46: *>
47: *> This is an auxiliary routine called by DSYTRD.
48: *> \endverbatim
49: *
50: * Arguments:
51: * ==========
52: *
53: *> \param[in] UPLO
54: *> \verbatim
55: *> UPLO is CHARACTER*1
56: *> Specifies whether the upper or lower triangular part of the
57: *> symmetric matrix A is stored:
58: *> = 'U': Upper triangular
59: *> = 'L': Lower triangular
60: *> \endverbatim
61: *>
62: *> \param[in] N
63: *> \verbatim
64: *> N is INTEGER
65: *> The order of the matrix A.
66: *> \endverbatim
67: *>
68: *> \param[in] NB
69: *> \verbatim
70: *> NB is INTEGER
71: *> The number of rows and columns to be reduced.
72: *> \endverbatim
73: *>
74: *> \param[in,out] A
75: *> \verbatim
76: *> A is DOUBLE PRECISION array, dimension (LDA,N)
77: *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
78: *> n-by-n upper triangular part of A contains the upper
79: *> triangular part of the matrix A, and the strictly lower
80: *> triangular part of A is not referenced. If UPLO = 'L', the
81: *> leading n-by-n lower triangular part of A contains the lower
82: *> triangular part of the matrix A, and the strictly upper
83: *> triangular part of A is not referenced.
84: *> On exit:
85: *> if UPLO = 'U', the last NB columns have been reduced to
86: *> tridiagonal form, with the diagonal elements overwriting
87: *> the diagonal elements of A; the elements above the diagonal
88: *> with the array TAU, represent the orthogonal matrix Q as a
89: *> product of elementary reflectors;
90: *> if UPLO = 'L', the first NB columns have been reduced to
91: *> tridiagonal form, with the diagonal elements overwriting
92: *> the diagonal elements of A; the elements below the diagonal
93: *> with the array TAU, represent the orthogonal matrix Q as a
94: *> product of elementary reflectors.
95: *> See Further Details.
96: *> \endverbatim
97: *>
98: *> \param[in] LDA
99: *> \verbatim
100: *> LDA is INTEGER
101: *> The leading dimension of the array A. LDA >= (1,N).
102: *> \endverbatim
103: *>
104: *> \param[out] E
105: *> \verbatim
106: *> E is DOUBLE PRECISION array, dimension (N-1)
107: *> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
108: *> elements of the last NB columns of the reduced matrix;
109: *> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
110: *> the first NB columns of the reduced matrix.
111: *> \endverbatim
112: *>
113: *> \param[out] TAU
114: *> \verbatim
115: *> TAU is DOUBLE PRECISION array, dimension (N-1)
116: *> The scalar factors of the elementary reflectors, stored in
117: *> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
118: *> See Further Details.
119: *> \endverbatim
120: *>
121: *> \param[out] W
122: *> \verbatim
123: *> W is DOUBLE PRECISION array, dimension (LDW,NB)
124: *> The n-by-nb matrix W required to update the unreduced part
125: *> of A.
126: *> \endverbatim
127: *>
128: *> \param[in] LDW
129: *> \verbatim
130: *> LDW is INTEGER
131: *> The leading dimension of the array W. LDW >= max(1,N).
132: *> \endverbatim
133: *
134: * Authors:
135: * ========
136: *
137: *> \author Univ. of Tennessee
138: *> \author Univ. of California Berkeley
139: *> \author Univ. of Colorado Denver
140: *> \author NAG Ltd.
141: *
142: *> \ingroup doubleOTHERauxiliary
143: *
144: *> \par Further Details:
145: * =====================
146: *>
147: *> \verbatim
148: *>
149: *> If UPLO = 'U', the matrix Q is represented as a product of elementary
150: *> reflectors
151: *>
152: *> Q = H(n) H(n-1) . . . H(n-nb+1).
153: *>
154: *> Each H(i) has the form
155: *>
156: *> H(i) = I - tau * v * v**T
157: *>
158: *> where tau is a real scalar, and v is a real vector with
159: *> v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
160: *> and tau in TAU(i-1).
161: *>
162: *> If UPLO = 'L', the matrix Q is represented as a product of elementary
163: *> reflectors
164: *>
165: *> Q = H(1) H(2) . . . H(nb).
166: *>
167: *> Each H(i) has the form
168: *>
169: *> H(i) = I - tau * v * v**T
170: *>
171: *> where tau is a real scalar, and v is a real vector with
172: *> v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
173: *> and tau in TAU(i).
174: *>
175: *> The elements of the vectors v together form the n-by-nb matrix V
176: *> which is needed, with W, to apply the transformation to the unreduced
177: *> part of the matrix, using a symmetric rank-2k update of the form:
178: *> A := A - V*W**T - W*V**T.
179: *>
180: *> The contents of A on exit are illustrated by the following examples
181: *> with n = 5 and nb = 2:
182: *>
183: *> if UPLO = 'U': if UPLO = 'L':
184: *>
185: *> ( a a a v4 v5 ) ( d )
186: *> ( a a v4 v5 ) ( 1 d )
187: *> ( a 1 v5 ) ( v1 1 a )
188: *> ( d 1 ) ( v1 v2 a a )
189: *> ( d ) ( v1 v2 a a a )
190: *>
191: *> where d denotes a diagonal element of the reduced matrix, a denotes
192: *> an element of the original matrix that is unchanged, and vi denotes
193: *> an element of the vector defining H(i).
194: *> \endverbatim
195: *>
196: * =====================================================================
197: SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
198: *
199: * -- LAPACK auxiliary routine --
200: * -- LAPACK is a software package provided by Univ. of Tennessee, --
201: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
202: *
203: * .. Scalar Arguments ..
204: CHARACTER UPLO
205: INTEGER LDA, LDW, N, NB
206: * ..
207: * .. Array Arguments ..
208: DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
209: * ..
210: *
211: * =====================================================================
212: *
213: * .. Parameters ..
214: DOUBLE PRECISION ZERO, ONE, HALF
215: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 )
216: * ..
217: * .. Local Scalars ..
218: INTEGER I, IW
219: DOUBLE PRECISION ALPHA
220: * ..
221: * .. External Subroutines ..
222: EXTERNAL DAXPY, DGEMV, DLARFG, DSCAL, DSYMV
223: * ..
224: * .. External Functions ..
225: LOGICAL LSAME
226: DOUBLE PRECISION DDOT
227: EXTERNAL LSAME, DDOT
228: * ..
229: * .. Intrinsic Functions ..
230: INTRINSIC MIN
231: * ..
232: * .. Executable Statements ..
233: *
234: * Quick return if possible
235: *
236: IF( N.LE.0 )
237: $ RETURN
238: *
239: IF( LSAME( UPLO, 'U' ) ) THEN
240: *
241: * Reduce last NB columns of upper triangle
242: *
243: DO 10 I = N, N - NB + 1, -1
244: IW = I - N + NB
245: IF( I.LT.N ) THEN
246: *
247: * Update A(1:i,i)
248: *
249: CALL DGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
250: $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
251: CALL DGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
252: $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
253: END IF
254: IF( I.GT.1 ) THEN
255: *
256: * Generate elementary reflector H(i) to annihilate
257: * A(1:i-2,i)
258: *
259: CALL DLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) )
260: E( I-1 ) = A( I-1, I )
261: A( I-1, I ) = ONE
262: *
263: * Compute W(1:i-1,i)
264: *
265: CALL DSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
266: $ ZERO, W( 1, IW ), 1 )
267: IF( I.LT.N ) THEN
268: CALL DGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ),
269: $ LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
270: CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
271: $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
272: $ W( 1, IW ), 1 )
273: CALL DGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ),
274: $ LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
275: CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
276: $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
277: $ W( 1, IW ), 1 )
278: END IF
279: CALL DSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
280: ALPHA = -HALF*TAU( I-1 )*DDOT( I-1, W( 1, IW ), 1,
281: $ A( 1, I ), 1 )
282: CALL DAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
283: END IF
284: *
285: 10 CONTINUE
286: ELSE
287: *
288: * Reduce first NB columns of lower triangle
289: *
290: DO 20 I = 1, NB
291: *
292: * Update A(i:n,i)
293: *
294: CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
295: $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
296: CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
297: $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
298: IF( I.LT.N ) THEN
299: *
300: * Generate elementary reflector H(i) to annihilate
301: * A(i+2:n,i)
302: *
303: CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
304: $ TAU( I ) )
305: E( I ) = A( I+1, I )
306: A( I+1, I ) = ONE
307: *
308: * Compute W(i+1:n,i)
309: *
310: CALL DSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
311: $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
312: CALL DGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW,
313: $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
314: CALL DGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),
315: $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
316: CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA,
317: $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
318: CALL DGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),
319: $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
320: CALL DSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
321: ALPHA = -HALF*TAU( I )*DDOT( N-I, W( I+1, I ), 1,
322: $ A( I+1, I ), 1 )
323: CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )
324: END IF
325: *
326: 20 CONTINUE
327: END IF
328: *
329: RETURN
330: *
331: * End of DLATRD
332: *
333: END
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