1: SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
2: $ LDY )
3: *
4: * -- LAPACK auxiliary routine (version 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
8: *
9: * .. Scalar Arguments ..
10: INTEGER LDA, LDX, LDY, M, N, NB
11: * ..
12: * .. Array Arguments ..
13: DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
14: $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
15: * ..
16: *
17: * Purpose
18: * =======
19: *
20: * DLABRD reduces the first NB rows and columns of a real general
21: * m by n matrix A to upper or lower bidiagonal form by an orthogonal
22: * transformation Q' * A * P, and returns the matrices X and Y which
23: * are needed to apply the transformation to the unreduced part of A.
24: *
25: * If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
26: * bidiagonal form.
27: *
28: * This is an auxiliary routine called by DGEBRD
29: *
30: * Arguments
31: * =========
32: *
33: * M (input) INTEGER
34: * The number of rows in the matrix A.
35: *
36: * N (input) INTEGER
37: * The number of columns in the matrix A.
38: *
39: * NB (input) INTEGER
40: * The number of leading rows and columns of A to be reduced.
41: *
42: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
43: * On entry, the m by n general matrix to be reduced.
44: * On exit, the first NB rows and columns of the matrix are
45: * overwritten; the rest of the array is unchanged.
46: * If m >= n, elements on and below the diagonal in the first NB
47: * columns, with the array TAUQ, represent the orthogonal
48: * matrix Q as a product of elementary reflectors; and
49: * elements above the diagonal in the first NB rows, with the
50: * array TAUP, represent the orthogonal matrix P as a product
51: * of elementary reflectors.
52: * If m < n, elements below the diagonal in the first NB
53: * columns, with the array TAUQ, represent the orthogonal
54: * matrix Q as a product of elementary reflectors, and
55: * elements on and above the diagonal in the first NB rows,
56: * with the array TAUP, represent the orthogonal matrix P as
57: * a product of elementary reflectors.
58: * See Further Details.
59: *
60: * LDA (input) INTEGER
61: * The leading dimension of the array A. LDA >= max(1,M).
62: *
63: * D (output) DOUBLE PRECISION array, dimension (NB)
64: * The diagonal elements of the first NB rows and columns of
65: * the reduced matrix. D(i) = A(i,i).
66: *
67: * E (output) DOUBLE PRECISION array, dimension (NB)
68: * The off-diagonal elements of the first NB rows and columns of
69: * the reduced matrix.
70: *
71: * TAUQ (output) DOUBLE PRECISION array dimension (NB)
72: * The scalar factors of the elementary reflectors which
73: * represent the orthogonal matrix Q. See Further Details.
74: *
75: * TAUP (output) DOUBLE PRECISION array, dimension (NB)
76: * The scalar factors of the elementary reflectors which
77: * represent the orthogonal matrix P. See Further Details.
78: *
79: * X (output) DOUBLE PRECISION array, dimension (LDX,NB)
80: * The m-by-nb matrix X required to update the unreduced part
81: * of A.
82: *
83: * LDX (input) INTEGER
84: * The leading dimension of the array X. LDX >= M.
85: *
86: * Y (output) DOUBLE PRECISION array, dimension (LDY,NB)
87: * The n-by-nb matrix Y required to update the unreduced part
88: * of A.
89: *
90: * LDY (input) INTEGER
91: * The leading dimension of the array Y. LDY >= N.
92: *
93: * Further Details
94: * ===============
95: *
96: * The matrices Q and P are represented as products of elementary
97: * reflectors:
98: *
99: * Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
100: *
101: * Each H(i) and G(i) has the form:
102: *
103: * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
104: *
105: * where tauq and taup are real scalars, and v and u are real vectors.
106: *
107: * If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
108: * A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
109: * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
110: *
111: * If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
112: * A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
113: * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
114: *
115: * The elements of the vectors v and u together form the m-by-nb matrix
116: * V and the nb-by-n matrix U' which are needed, with X and Y, to apply
117: * the transformation to the unreduced part of the matrix, using a block
118: * update of the form: A := A - V*Y' - X*U'.
119: *
120: * The contents of A on exit are illustrated by the following examples
121: * with nb = 2:
122: *
123: * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
124: *
125: * ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
126: * ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
127: * ( v1 v2 a a a ) ( v1 1 a a a a )
128: * ( v1 v2 a a a ) ( v1 v2 a a a a )
129: * ( v1 v2 a a a ) ( v1 v2 a a a a )
130: * ( v1 v2 a a a )
131: *
132: * where a denotes an element of the original matrix which is unchanged,
133: * vi denotes an element of the vector defining H(i), and ui an element
134: * of the vector defining G(i).
135: *
136: * =====================================================================
137: *
138: * .. Parameters ..
139: DOUBLE PRECISION ZERO, ONE
140: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
141: * ..
142: * .. Local Scalars ..
143: INTEGER I
144: * ..
145: * .. External Subroutines ..
146: EXTERNAL DGEMV, DLARFG, DSCAL
147: * ..
148: * .. Intrinsic Functions ..
149: INTRINSIC MIN
150: * ..
151: * .. Executable Statements ..
152: *
153: * Quick return if possible
154: *
155: IF( M.LE.0 .OR. N.LE.0 )
156: $ RETURN
157: *
158: IF( M.GE.N ) THEN
159: *
160: * Reduce to upper bidiagonal form
161: *
162: DO 10 I = 1, NB
163: *
164: * Update A(i:m,i)
165: *
166: CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
167: $ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
168: CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
169: $ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
170: *
171: * Generate reflection Q(i) to annihilate A(i+1:m,i)
172: *
173: CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
174: $ TAUQ( I ) )
175: D( I ) = A( I, I )
176: IF( I.LT.N ) THEN
177: A( I, I ) = ONE
178: *
179: * Compute Y(i+1:n,i)
180: *
181: CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ),
182: $ LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )
183: CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA,
184: $ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
185: CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
186: $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
187: CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX,
188: $ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
189: CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
190: $ LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
191: CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
192: *
193: * Update A(i,i+1:n)
194: *
195: CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
196: $ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
197: CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
198: $ LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )
199: *
200: * Generate reflection P(i) to annihilate A(i,i+2:n)
201: *
202: CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
203: $ LDA, TAUP( I ) )
204: E( I ) = A( I, I+1 )
205: A( I, I+1 ) = ONE
206: *
207: * Compute X(i+1:m,i)
208: *
209: CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),
210: $ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
211: CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY,
212: $ A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
213: CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
214: $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
215: CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),
216: $ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
217: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
218: $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
219: CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
220: END IF
221: 10 CONTINUE
222: ELSE
223: *
224: * Reduce to lower bidiagonal form
225: *
226: DO 20 I = 1, NB
227: *
228: * Update A(i,i:n)
229: *
230: CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
231: $ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
232: CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA,
233: $ X( I, 1 ), LDX, ONE, A( I, I ), LDA )
234: *
235: * Generate reflection P(i) to annihilate A(i,i+1:n)
236: *
237: CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
238: $ TAUP( I ) )
239: D( I ) = A( I, I )
240: IF( I.LT.M ) THEN
241: A( I, I ) = ONE
242: *
243: * Compute X(i+1:m,i)
244: *
245: CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),
246: $ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
247: CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY,
248: $ A( I, I ), LDA, ZERO, X( 1, I ), 1 )
249: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
250: $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
251: CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),
252: $ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
253: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
254: $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
255: CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
256: *
257: * Update A(i+1:m,i)
258: *
259: CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
260: $ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
261: CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
262: $ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
263: *
264: * Generate reflection Q(i) to annihilate A(i+2:m,i)
265: *
266: CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
267: $ TAUQ( I ) )
268: E( I ) = A( I+1, I )
269: A( I+1, I ) = ONE
270: *
271: * Compute Y(i+1:n,i)
272: *
273: CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ),
274: $ LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )
275: CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA,
276: $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
277: CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
278: $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
279: CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX,
280: $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
281: CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA,
282: $ Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
283: CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
284: END IF
285: 20 CONTINUE
286: END IF
287: RETURN
288: *
289: * End of DLABRD
290: *
291: END
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