1: *> \brief \b DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLABRD + dependencies
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlabrd.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
22: * LDY )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER LDA, LDX, LDY, M, N, NB
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
29: * $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DLABRD reduces the first NB rows and columns of a real general
39: *> m by n matrix A to upper or lower bidiagonal form by an orthogonal
40: *> transformation Q**T * A * P, and returns the matrices X and Y which
41: *> are needed to apply the transformation to the unreduced part of A.
42: *>
43: *> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
44: *> bidiagonal form.
45: *>
46: *> This is an auxiliary routine called by DGEBRD
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] M
53: *> \verbatim
54: *> M is INTEGER
55: *> The number of rows in the matrix A.
56: *> \endverbatim
57: *>
58: *> \param[in] N
59: *> \verbatim
60: *> N is INTEGER
61: *> The number of columns in the matrix A.
62: *> \endverbatim
63: *>
64: *> \param[in] NB
65: *> \verbatim
66: *> NB is INTEGER
67: *> The number of leading rows and columns of A to be reduced.
68: *> \endverbatim
69: *>
70: *> \param[in,out] A
71: *> \verbatim
72: *> A is DOUBLE PRECISION array, dimension (LDA,N)
73: *> On entry, the m by n general matrix to be reduced.
74: *> On exit, the first NB rows and columns of the matrix are
75: *> overwritten; the rest of the array is unchanged.
76: *> If m >= n, elements on and below the diagonal in the first NB
77: *> columns, with the array TAUQ, represent the orthogonal
78: *> matrix Q as a product of elementary reflectors; and
79: *> elements above the diagonal in the first NB rows, with the
80: *> array TAUP, represent the orthogonal matrix P as a product
81: *> of elementary reflectors.
82: *> If m < n, elements below the diagonal in the first NB
83: *> columns, with the array TAUQ, represent the orthogonal
84: *> matrix Q as a product of elementary reflectors, and
85: *> elements on and above the diagonal in the first NB rows,
86: *> with the array TAUP, represent the orthogonal matrix P as
87: *> a product of elementary reflectors.
88: *> See Further Details.
89: *> \endverbatim
90: *>
91: *> \param[in] LDA
92: *> \verbatim
93: *> LDA is INTEGER
94: *> The leading dimension of the array A. LDA >= max(1,M).
95: *> \endverbatim
96: *>
97: *> \param[out] D
98: *> \verbatim
99: *> D is DOUBLE PRECISION array, dimension (NB)
100: *> The diagonal elements of the first NB rows and columns of
101: *> the reduced matrix. D(i) = A(i,i).
102: *> \endverbatim
103: *>
104: *> \param[out] E
105: *> \verbatim
106: *> E is DOUBLE PRECISION array, dimension (NB)
107: *> The off-diagonal elements of the first NB rows and columns of
108: *> the reduced matrix.
109: *> \endverbatim
110: *>
111: *> \param[out] TAUQ
112: *> \verbatim
113: *> TAUQ is DOUBLE PRECISION array dimension (NB)
114: *> The scalar factors of the elementary reflectors which
115: *> represent the orthogonal matrix Q. See Further Details.
116: *> \endverbatim
117: *>
118: *> \param[out] TAUP
119: *> \verbatim
120: *> TAUP is DOUBLE PRECISION array, dimension (NB)
121: *> The scalar factors of the elementary reflectors which
122: *> represent the orthogonal matrix P. See Further Details.
123: *> \endverbatim
124: *>
125: *> \param[out] X
126: *> \verbatim
127: *> X is DOUBLE PRECISION array, dimension (LDX,NB)
128: *> The m-by-nb matrix X required to update the unreduced part
129: *> of A.
130: *> \endverbatim
131: *>
132: *> \param[in] LDX
133: *> \verbatim
134: *> LDX is INTEGER
135: *> The leading dimension of the array X. LDX >= max(1,M).
136: *> \endverbatim
137: *>
138: *> \param[out] Y
139: *> \verbatim
140: *> Y is DOUBLE PRECISION array, dimension (LDY,NB)
141: *> The n-by-nb matrix Y required to update the unreduced part
142: *> of A.
143: *> \endverbatim
144: *>
145: *> \param[in] LDY
146: *> \verbatim
147: *> LDY is INTEGER
148: *> The leading dimension of the array Y. LDY >= max(1,N).
149: *> \endverbatim
150: *
151: * Authors:
152: * ========
153: *
154: *> \author Univ. of Tennessee
155: *> \author Univ. of California Berkeley
156: *> \author Univ. of Colorado Denver
157: *> \author NAG Ltd.
158: *
159: *> \date September 2012
160: *
161: *> \ingroup doubleOTHERauxiliary
162: *
163: *> \par Further Details:
164: * =====================
165: *>
166: *> \verbatim
167: *>
168: *> The matrices Q and P are represented as products of elementary
169: *> reflectors:
170: *>
171: *> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
172: *>
173: *> Each H(i) and G(i) has the form:
174: *>
175: *> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
176: *>
177: *> where tauq and taup are real scalars, and v and u are real vectors.
178: *>
179: *> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
180: *> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
181: *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
182: *>
183: *> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
184: *> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
185: *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
186: *>
187: *> The elements of the vectors v and u together form the m-by-nb matrix
188: *> V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
189: *> the transformation to the unreduced part of the matrix, using a block
190: *> update of the form: A := A - V*Y**T - X*U**T.
191: *>
192: *> The contents of A on exit are illustrated by the following examples
193: *> with nb = 2:
194: *>
195: *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
196: *>
197: *> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
198: *> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
199: *> ( v1 v2 a a a ) ( v1 1 a a a a )
200: *> ( v1 v2 a a a ) ( v1 v2 a a a a )
201: *> ( v1 v2 a a a ) ( v1 v2 a a a a )
202: *> ( v1 v2 a a a )
203: *>
204: *> where a denotes an element of the original matrix which is unchanged,
205: *> vi denotes an element of the vector defining H(i), and ui an element
206: *> of the vector defining G(i).
207: *> \endverbatim
208: *>
209: * =====================================================================
210: SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
211: $ LDY )
212: *
213: * -- LAPACK auxiliary routine (version 3.4.2) --
214: * -- LAPACK is a software package provided by Univ. of Tennessee, --
215: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216: * September 2012
217: *
218: * .. Scalar Arguments ..
219: INTEGER LDA, LDX, LDY, M, N, NB
220: * ..
221: * .. Array Arguments ..
222: DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
223: $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
224: * ..
225: *
226: * =====================================================================
227: *
228: * .. Parameters ..
229: DOUBLE PRECISION ZERO, ONE
230: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
231: * ..
232: * .. Local Scalars ..
233: INTEGER I
234: * ..
235: * .. External Subroutines ..
236: EXTERNAL DGEMV, DLARFG, DSCAL
237: * ..
238: * .. Intrinsic Functions ..
239: INTRINSIC MIN
240: * ..
241: * .. Executable Statements ..
242: *
243: * Quick return if possible
244: *
245: IF( M.LE.0 .OR. N.LE.0 )
246: $ RETURN
247: *
248: IF( M.GE.N ) THEN
249: *
250: * Reduce to upper bidiagonal form
251: *
252: DO 10 I = 1, NB
253: *
254: * Update A(i:m,i)
255: *
256: CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
257: $ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
258: CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
259: $ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
260: *
261: * Generate reflection Q(i) to annihilate A(i+1:m,i)
262: *
263: CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
264: $ TAUQ( I ) )
265: D( I ) = A( I, I )
266: IF( I.LT.N ) THEN
267: A( I, I ) = ONE
268: *
269: * Compute Y(i+1:n,i)
270: *
271: CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ),
272: $ LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )
273: CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA,
274: $ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
275: CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
276: $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
277: CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX,
278: $ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
279: CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
280: $ LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
281: CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
282: *
283: * Update A(i,i+1:n)
284: *
285: CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
286: $ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
287: CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
288: $ LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )
289: *
290: * Generate reflection P(i) to annihilate A(i,i+2:n)
291: *
292: CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
293: $ LDA, TAUP( I ) )
294: E( I ) = A( I, I+1 )
295: A( I, I+1 ) = ONE
296: *
297: * Compute X(i+1:m,i)
298: *
299: CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),
300: $ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
301: CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY,
302: $ A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
303: CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
304: $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
305: CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),
306: $ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
307: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
308: $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
309: CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
310: END IF
311: 10 CONTINUE
312: ELSE
313: *
314: * Reduce to lower bidiagonal form
315: *
316: DO 20 I = 1, NB
317: *
318: * Update A(i,i:n)
319: *
320: CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
321: $ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
322: CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA,
323: $ X( I, 1 ), LDX, ONE, A( I, I ), LDA )
324: *
325: * Generate reflection P(i) to annihilate A(i,i+1:n)
326: *
327: CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
328: $ TAUP( I ) )
329: D( I ) = A( I, I )
330: IF( I.LT.M ) THEN
331: A( I, I ) = ONE
332: *
333: * Compute X(i+1:m,i)
334: *
335: CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),
336: $ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
337: CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY,
338: $ A( I, I ), LDA, ZERO, X( 1, I ), 1 )
339: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
340: $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
341: CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),
342: $ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
343: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
344: $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
345: CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
346: *
347: * Update A(i+1:m,i)
348: *
349: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
350: $ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
351: CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
352: $ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
353: *
354: * Generate reflection Q(i) to annihilate A(i+2:m,i)
355: *
356: CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
357: $ TAUQ( I ) )
358: E( I ) = A( I+1, I )
359: A( I+1, I ) = ONE
360: *
361: * Compute Y(i+1:n,i)
362: *
363: CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ),
364: $ LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )
365: CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA,
366: $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
367: CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
368: $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
369: CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX,
370: $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
371: CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA,
372: $ Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
373: CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
374: END IF
375: 20 CONTINUE
376: END IF
377: RETURN
378: *
379: * End of DLABRD
380: *
381: END
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