Annotation of rpl/lapack/lapack/dlabrd.f, revision 1.20
1.12 bertrand 1: *> \brief \b DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
1.9 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
1.16 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.9 bertrand 7: *
8: *> \htmlonly
1.16 bertrand 9: *> Download DLABRD + dependencies
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11: *> [TGZ]</a>
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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">
1.9 bertrand 15: *> [TXT]</a>
1.16 bertrand 16: *> \endhtmlonly
1.9 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
22: * LDY )
1.16 bertrand 23: *
1.9 bertrand 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: * ..
1.16 bertrand 31: *
1.9 bertrand 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
1.18 bertrand 113: *> TAUQ is DOUBLE PRECISION array, dimension (NB)
1.9 bertrand 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: *
1.16 bertrand 154: *> \author Univ. of Tennessee
155: *> \author Univ. of California Berkeley
156: *> \author Univ. of Colorado Denver
157: *> \author NAG Ltd.
1.9 bertrand 158: *
159: *> \ingroup doubleOTHERauxiliary
160: *
161: *> \par Further Details:
162: * =====================
163: *>
164: *> \verbatim
165: *>
166: *> The matrices Q and P are represented as products of elementary
167: *> reflectors:
168: *>
169: *> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
170: *>
171: *> Each H(i) and G(i) has the form:
172: *>
173: *> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
174: *>
175: *> where tauq and taup are real scalars, and v and u are real vectors.
176: *>
177: *> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
178: *> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
179: *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
180: *>
181: *> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
182: *> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
183: *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
184: *>
185: *> The elements of the vectors v and u together form the m-by-nb matrix
186: *> V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
187: *> the transformation to the unreduced part of the matrix, using a block
188: *> update of the form: A := A - V*Y**T - X*U**T.
189: *>
190: *> The contents of A on exit are illustrated by the following examples
191: *> with nb = 2:
192: *>
193: *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
194: *>
195: *> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
196: *> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
197: *> ( v1 v2 a a a ) ( v1 1 a a a a )
198: *> ( v1 v2 a a a ) ( v1 v2 a a a a )
199: *> ( v1 v2 a a a ) ( v1 v2 a a a a )
200: *> ( v1 v2 a a a )
201: *>
202: *> where a denotes an element of the original matrix which is unchanged,
203: *> vi denotes an element of the vector defining H(i), and ui an element
204: *> of the vector defining G(i).
205: *> \endverbatim
206: *>
207: * =====================================================================
1.1 bertrand 208: SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
209: $ LDY )
210: *
1.20 ! bertrand 211: * -- LAPACK auxiliary routine --
1.1 bertrand 212: * -- LAPACK is a software package provided by Univ. of Tennessee, --
213: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
214: *
215: * .. Scalar Arguments ..
216: INTEGER LDA, LDX, LDY, M, N, NB
217: * ..
218: * .. Array Arguments ..
219: DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
220: $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
221: * ..
222: *
223: * =====================================================================
224: *
225: * .. Parameters ..
226: DOUBLE PRECISION ZERO, ONE
227: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
228: * ..
229: * .. Local Scalars ..
230: INTEGER I
231: * ..
232: * .. External Subroutines ..
233: EXTERNAL DGEMV, DLARFG, DSCAL
234: * ..
235: * .. Intrinsic Functions ..
236: INTRINSIC MIN
237: * ..
238: * .. Executable Statements ..
239: *
240: * Quick return if possible
241: *
242: IF( M.LE.0 .OR. N.LE.0 )
243: $ RETURN
244: *
245: IF( M.GE.N ) THEN
246: *
247: * Reduce to upper bidiagonal form
248: *
249: DO 10 I = 1, NB
250: *
251: * Update A(i:m,i)
252: *
253: CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
254: $ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
255: CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
256: $ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
257: *
258: * Generate reflection Q(i) to annihilate A(i+1:m,i)
259: *
260: CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
261: $ TAUQ( I ) )
262: D( I ) = A( I, I )
263: IF( I.LT.N ) THEN
264: A( I, I ) = ONE
265: *
266: * Compute Y(i+1:n,i)
267: *
268: CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ),
269: $ LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )
270: CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA,
271: $ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
272: CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
273: $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
274: CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX,
275: $ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
276: CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
277: $ LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
278: CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
279: *
280: * Update A(i,i+1:n)
281: *
282: CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
283: $ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
284: CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
285: $ LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )
286: *
287: * Generate reflection P(i) to annihilate A(i,i+2:n)
288: *
289: CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
290: $ LDA, TAUP( I ) )
291: E( I ) = A( I, I+1 )
292: A( I, I+1 ) = ONE
293: *
294: * Compute X(i+1:m,i)
295: *
296: CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),
297: $ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
298: CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY,
299: $ A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
300: CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
301: $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
302: CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),
303: $ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
304: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
305: $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
306: CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
307: END IF
308: 10 CONTINUE
309: ELSE
310: *
311: * Reduce to lower bidiagonal form
312: *
313: DO 20 I = 1, NB
314: *
315: * Update A(i,i:n)
316: *
317: CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
318: $ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
319: CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA,
320: $ X( I, 1 ), LDX, ONE, A( I, I ), LDA )
321: *
322: * Generate reflection P(i) to annihilate A(i,i+1:n)
323: *
324: CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
325: $ TAUP( I ) )
326: D( I ) = A( I, I )
327: IF( I.LT.M ) THEN
328: A( I, I ) = ONE
329: *
330: * Compute X(i+1:m,i)
331: *
332: CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),
333: $ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
334: CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY,
335: $ A( I, I ), LDA, ZERO, X( 1, I ), 1 )
336: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
337: $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
338: CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),
339: $ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
340: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
341: $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
342: CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
343: *
344: * Update A(i+1:m,i)
345: *
346: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
347: $ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
348: CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
349: $ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
350: *
351: * Generate reflection Q(i) to annihilate A(i+2:m,i)
352: *
353: CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
354: $ TAUQ( I ) )
355: E( I ) = A( I+1, I )
356: A( I+1, I ) = ONE
357: *
358: * Compute Y(i+1:n,i)
359: *
360: CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ),
361: $ LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )
362: CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA,
363: $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
364: CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
365: $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
366: CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX,
367: $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
368: CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA,
369: $ Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
370: CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
371: END IF
372: 20 CONTINUE
373: END IF
374: RETURN
375: *
376: * End of DLABRD
377: *
378: END
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