1: *> \brief \b DGEQPF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, M, N
25: * ..
26: * .. Array Arguments ..
27: * INTEGER JPVT( * )
28: * DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> This routine is deprecated and has been replaced by routine DGEQP3.
38: *>
39: *> DGEQPF computes a QR factorization with column pivoting of a
40: *> real M-by-N matrix A: A*P = Q*R.
41: *> \endverbatim
42: *
43: * Arguments:
44: * ==========
45: *
46: *> \param[in] M
47: *> \verbatim
48: *> M is INTEGER
49: *> The number of rows of the matrix A. M >= 0.
50: *> \endverbatim
51: *>
52: *> \param[in] N
53: *> \verbatim
54: *> N is INTEGER
55: *> The number of columns of the matrix A. N >= 0
56: *> \endverbatim
57: *>
58: *> \param[in,out] A
59: *> \verbatim
60: *> A is DOUBLE PRECISION array, dimension (LDA,N)
61: *> On entry, the M-by-N matrix A.
62: *> On exit, the upper triangle of the array contains the
63: *> min(M,N)-by-N upper triangular matrix R; the elements
64: *> below the diagonal, together with the array TAU,
65: *> represent the orthogonal matrix Q as a product of
66: *> min(m,n) elementary reflectors.
67: *> \endverbatim
68: *>
69: *> \param[in] LDA
70: *> \verbatim
71: *> LDA is INTEGER
72: *> The leading dimension of the array A. LDA >= max(1,M).
73: *> \endverbatim
74: *>
75: *> \param[in,out] JPVT
76: *> \verbatim
77: *> JPVT is INTEGER array, dimension (N)
78: *> On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
79: *> to the front of A*P (a leading column); if JPVT(i) = 0,
80: *> the i-th column of A is a free column.
81: *> On exit, if JPVT(i) = k, then the i-th column of A*P
82: *> was the k-th column of A.
83: *> \endverbatim
84: *>
85: *> \param[out] TAU
86: *> \verbatim
87: *> TAU is DOUBLE PRECISION array, dimension (min(M,N))
88: *> The scalar factors of the elementary reflectors.
89: *> \endverbatim
90: *>
91: *> \param[out] WORK
92: *> \verbatim
93: *> WORK is DOUBLE PRECISION array, dimension (3*N)
94: *> \endverbatim
95: *>
96: *> \param[out] INFO
97: *> \verbatim
98: *> INFO is INTEGER
99: *> = 0: successful exit
100: *> < 0: if INFO = -i, the i-th argument had an illegal value
101: *> \endverbatim
102: *
103: * Authors:
104: * ========
105: *
106: *> \author Univ. of Tennessee
107: *> \author Univ. of California Berkeley
108: *> \author Univ. of Colorado Denver
109: *> \author NAG Ltd.
110: *
111: *> \ingroup doubleGEcomputational
112: *
113: *> \par Further Details:
114: * =====================
115: *>
116: *> \verbatim
117: *>
118: *> The matrix Q is represented as a product of elementary reflectors
119: *>
120: *> Q = H(1) H(2) . . . H(n)
121: *>
122: *> Each H(i) has the form
123: *>
124: *> H = I - tau * v * v**T
125: *>
126: *> where tau is a real scalar, and v is a real vector with
127: *> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).
128: *>
129: *> The matrix P is represented in jpvt as follows: If
130: *> jpvt(j) = i
131: *> then the jth column of P is the ith canonical unit vector.
132: *>
133: *> Partial column norm updating strategy modified by
134: *> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
135: *> University of Zagreb, Croatia.
136: *> -- April 2011 --
137: *> For more details see LAPACK Working Note 176.
138: *> \endverbatim
139: *>
140: * =====================================================================
141: SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
142: *
143: * -- LAPACK computational routine --
144: * -- LAPACK is a software package provided by Univ. of Tennessee, --
145: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146: *
147: * .. Scalar Arguments ..
148: INTEGER INFO, LDA, M, N
149: * ..
150: * .. Array Arguments ..
151: INTEGER JPVT( * )
152: DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
153: * ..
154: *
155: * =====================================================================
156: *
157: * .. Parameters ..
158: DOUBLE PRECISION ZERO, ONE
159: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
160: * ..
161: * .. Local Scalars ..
162: INTEGER I, ITEMP, J, MA, MN, PVT
163: DOUBLE PRECISION AII, TEMP, TEMP2, TOL3Z
164: * ..
165: * .. External Subroutines ..
166: EXTERNAL DGEQR2, DLARF, DLARFG, DORM2R, DSWAP, XERBLA
167: * ..
168: * .. Intrinsic Functions ..
169: INTRINSIC ABS, MAX, MIN, SQRT
170: * ..
171: * .. External Functions ..
172: INTEGER IDAMAX
173: DOUBLE PRECISION DLAMCH, DNRM2
174: EXTERNAL IDAMAX, DLAMCH, DNRM2
175: * ..
176: * .. Executable Statements ..
177: *
178: * Test the input arguments
179: *
180: INFO = 0
181: IF( M.LT.0 ) THEN
182: INFO = -1
183: ELSE IF( N.LT.0 ) THEN
184: INFO = -2
185: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
186: INFO = -4
187: END IF
188: IF( INFO.NE.0 ) THEN
189: CALL XERBLA( 'DGEQPF', -INFO )
190: RETURN
191: END IF
192: *
193: MN = MIN( M, N )
194: TOL3Z = SQRT(DLAMCH('Epsilon'))
195: *
196: * Move initial columns up front
197: *
198: ITEMP = 1
199: DO 10 I = 1, N
200: IF( JPVT( I ).NE.0 ) THEN
201: IF( I.NE.ITEMP ) THEN
202: CALL DSWAP( M, A( 1, I ), 1, A( 1, ITEMP ), 1 )
203: JPVT( I ) = JPVT( ITEMP )
204: JPVT( ITEMP ) = I
205: ELSE
206: JPVT( I ) = I
207: END IF
208: ITEMP = ITEMP + 1
209: ELSE
210: JPVT( I ) = I
211: END IF
212: 10 CONTINUE
213: ITEMP = ITEMP - 1
214: *
215: * Compute the QR factorization and update remaining columns
216: *
217: IF( ITEMP.GT.0 ) THEN
218: MA = MIN( ITEMP, M )
219: CALL DGEQR2( M, MA, A, LDA, TAU, WORK, INFO )
220: IF( MA.LT.N ) THEN
221: CALL DORM2R( 'Left', 'Transpose', M, N-MA, MA, A, LDA, TAU,
222: $ A( 1, MA+1 ), LDA, WORK, INFO )
223: END IF
224: END IF
225: *
226: IF( ITEMP.LT.MN ) THEN
227: *
228: * Initialize partial column norms. The first n elements of
229: * work store the exact column norms.
230: *
231: DO 20 I = ITEMP + 1, N
232: WORK( I ) = DNRM2( M-ITEMP, A( ITEMP+1, I ), 1 )
233: WORK( N+I ) = WORK( I )
234: 20 CONTINUE
235: *
236: * Compute factorization
237: *
238: DO 40 I = ITEMP + 1, MN
239: *
240: * Determine ith pivot column and swap if necessary
241: *
242: PVT = ( I-1 ) + IDAMAX( N-I+1, WORK( I ), 1 )
243: *
244: IF( PVT.NE.I ) THEN
245: CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
246: ITEMP = JPVT( PVT )
247: JPVT( PVT ) = JPVT( I )
248: JPVT( I ) = ITEMP
249: WORK( PVT ) = WORK( I )
250: WORK( N+PVT ) = WORK( N+I )
251: END IF
252: *
253: * Generate elementary reflector H(i)
254: *
255: IF( I.LT.M ) THEN
256: CALL DLARFG( M-I+1, A( I, I ), A( I+1, I ), 1, TAU( I ) )
257: ELSE
258: CALL DLARFG( 1, A( M, M ), A( M, M ), 1, TAU( M ) )
259: END IF
260: *
261: IF( I.LT.N ) THEN
262: *
263: * Apply H(i) to A(i:m,i+1:n) from the left
264: *
265: AII = A( I, I )
266: A( I, I ) = ONE
267: CALL DLARF( 'LEFT', M-I+1, N-I, A( I, I ), 1, TAU( I ),
268: $ A( I, I+1 ), LDA, WORK( 2*N+1 ) )
269: A( I, I ) = AII
270: END IF
271: *
272: * Update partial column norms
273: *
274: DO 30 J = I + 1, N
275: IF( WORK( J ).NE.ZERO ) THEN
276: *
277: * NOTE: The following 4 lines follow from the analysis in
278: * Lapack Working Note 176.
279: *
280: TEMP = ABS( A( I, J ) ) / WORK( J )
281: TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
282: TEMP2 = TEMP*( WORK( J ) / WORK( N+J ) )**2
283: IF( TEMP2 .LE. TOL3Z ) THEN
284: IF( M-I.GT.0 ) THEN
285: WORK( J ) = DNRM2( M-I, A( I+1, J ), 1 )
286: WORK( N+J ) = WORK( J )
287: ELSE
288: WORK( J ) = ZERO
289: WORK( N+J ) = ZERO
290: END IF
291: ELSE
292: WORK( J ) = WORK( J )*SQRT( TEMP )
293: END IF
294: END IF
295: 30 CONTINUE
296: *
297: 40 CONTINUE
298: END IF
299: RETURN
300: *
301: * End of DGEQPF
302: *
303: END
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