1: *> \brief \b ZGEQRF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGEQRF + 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/zgeqrf.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, LWORK, M, N
25: * ..
26: * .. Array Arguments ..
27: * COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
28: * ..
29: *
30: *
31: *> \par Purpose:
32: * =============
33: *>
34: *> \verbatim
35: *>
36: *> ZGEQRF computes a QR factorization of a complex M-by-N matrix A:
37: *>
38: *> A = Q * ( R ),
39: *> ( 0 )
40: *>
41: *> where:
42: *>
43: *> Q is a M-by-M orthogonal matrix;
44: *> R is an upper-triangular N-by-N matrix;
45: *> 0 is a (M-N)-by-N zero matrix, if M > N.
46: *>
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] M
53: *> \verbatim
54: *> M is INTEGER
55: *> The number of rows of the matrix A. M >= 0.
56: *> \endverbatim
57: *>
58: *> \param[in] N
59: *> \verbatim
60: *> N is INTEGER
61: *> The number of columns of the matrix A. N >= 0.
62: *> \endverbatim
63: *>
64: *> \param[in,out] A
65: *> \verbatim
66: *> A is COMPLEX*16 array, dimension (LDA,N)
67: *> On entry, the M-by-N matrix A.
68: *> On exit, the elements on and above the diagonal of the array
69: *> contain the min(M,N)-by-N upper trapezoidal matrix R (R is
70: *> upper triangular if m >= n); the elements below the diagonal,
71: *> with the array TAU, represent the unitary matrix Q as a
72: *> product of min(m,n) elementary reflectors (see Further
73: *> Details).
74: *> \endverbatim
75: *>
76: *> \param[in] LDA
77: *> \verbatim
78: *> LDA is INTEGER
79: *> The leading dimension of the array A. LDA >= max(1,M).
80: *> \endverbatim
81: *>
82: *> \param[out] TAU
83: *> \verbatim
84: *> TAU is COMPLEX*16 array, dimension (min(M,N))
85: *> The scalar factors of the elementary reflectors (see Further
86: *> Details).
87: *> \endverbatim
88: *>
89: *> \param[out] WORK
90: *> \verbatim
91: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
92: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
93: *> \endverbatim
94: *>
95: *> \param[in] LWORK
96: *> \verbatim
97: *> LWORK is INTEGER
98: *> The dimension of the array WORK.
99: *> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
100: *> For optimum performance LWORK >= N*NB, where NB is
101: *> the optimal blocksize.
102: *>
103: *> If LWORK = -1, then a workspace query is assumed; the routine
104: *> only calculates the optimal size of the WORK array, returns
105: *> this value as the first entry of the WORK array, and no error
106: *> message related to LWORK is issued by XERBLA.
107: *> \endverbatim
108: *>
109: *> \param[out] INFO
110: *> \verbatim
111: *> INFO is INTEGER
112: *> = 0: successful exit
113: *> < 0: if INFO = -i, the i-th argument had an illegal value
114: *> \endverbatim
115: *
116: * Authors:
117: * ========
118: *
119: *> \author Univ. of Tennessee
120: *> \author Univ. of California Berkeley
121: *> \author Univ. of Colorado Denver
122: *> \author NAG Ltd.
123: *
124: *> \ingroup complex16GEcomputational
125: *
126: *> \par Further Details:
127: * =====================
128: *>
129: *> \verbatim
130: *>
131: *> The matrix Q is represented as a product of elementary reflectors
132: *>
133: *> Q = H(1) H(2) . . . H(k), where k = min(m,n).
134: *>
135: *> Each H(i) has the form
136: *>
137: *> H(i) = I - tau * v * v**H
138: *>
139: *> where tau is a complex scalar, and v is a complex vector with
140: *> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
141: *> and tau in TAU(i).
142: *> \endverbatim
143: *>
144: * =====================================================================
145: SUBROUTINE ZGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
146: *
147: * -- LAPACK computational routine --
148: * -- LAPACK is a software package provided by Univ. of Tennessee, --
149: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
150: *
151: * .. Scalar Arguments ..
152: INTEGER INFO, LDA, LWORK, M, N
153: * ..
154: * .. Array Arguments ..
155: COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
156: * ..
157: *
158: * =====================================================================
159: *
160: * .. Local Scalars ..
161: LOGICAL LQUERY
162: INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
163: $ NBMIN, NX
164: * ..
165: * .. External Subroutines ..
166: EXTERNAL XERBLA, ZGEQR2, ZLARFB, ZLARFT
167: * ..
168: * .. Intrinsic Functions ..
169: INTRINSIC MAX, MIN
170: * ..
171: * .. External Functions ..
172: INTEGER ILAENV
173: EXTERNAL ILAENV
174: * ..
175: * .. Executable Statements ..
176: *
177: * Test the input arguments
178: *
179: K = MIN( M, N )
180: INFO = 0
181: NB = ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
182: LQUERY = ( LWORK.EQ.-1 )
183: IF( M.LT.0 ) THEN
184: INFO = -1
185: ELSE IF( N.LT.0 ) THEN
186: INFO = -2
187: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
188: INFO = -4
189: ELSE IF( .NOT.LQUERY ) THEN
190: IF( LWORK.LE.0 .OR. ( M.GT.0 .AND. LWORK.LT.MAX( 1, N ) ) )
191: $ INFO = -7
192: END IF
193: IF( INFO.NE.0 ) THEN
194: CALL XERBLA( 'ZGEQRF', -INFO )
195: RETURN
196: ELSE IF( LQUERY ) THEN
197: IF( K.EQ.0 ) THEN
198: LWKOPT = 1
199: ELSE
200: LWKOPT = N*NB
201: END IF
202: WORK( 1 ) = LWKOPT
203: RETURN
204: END IF
205: *
206: * Quick return if possible
207: *
208: IF( K.EQ.0 ) THEN
209: WORK( 1 ) = 1
210: RETURN
211: END IF
212: *
213: NBMIN = 2
214: NX = 0
215: IWS = N
216: IF( NB.GT.1 .AND. NB.LT.K ) THEN
217: *
218: * Determine when to cross over from blocked to unblocked code.
219: *
220: NX = MAX( 0, ILAENV( 3, 'ZGEQRF', ' ', M, N, -1, -1 ) )
221: IF( NX.LT.K ) THEN
222: *
223: * Determine if workspace is large enough for blocked code.
224: *
225: LDWORK = N
226: IWS = LDWORK*NB
227: IF( LWORK.LT.IWS ) THEN
228: *
229: * Not enough workspace to use optimal NB: reduce NB and
230: * determine the minimum value of NB.
231: *
232: NB = LWORK / LDWORK
233: NBMIN = MAX( 2, ILAENV( 2, 'ZGEQRF', ' ', M, N, -1,
234: $ -1 ) )
235: END IF
236: END IF
237: END IF
238: *
239: IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
240: *
241: * Use blocked code initially
242: *
243: DO 10 I = 1, K - NX, NB
244: IB = MIN( K-I+1, NB )
245: *
246: * Compute the QR factorization of the current block
247: * A(i:m,i:i+ib-1)
248: *
249: CALL ZGEQR2( M-I+1, IB, A( I, I ), LDA, TAU( I ), WORK,
250: $ IINFO )
251: IF( I+IB.LE.N ) THEN
252: *
253: * Form the triangular factor of the block reflector
254: * H = H(i) H(i+1) . . . H(i+ib-1)
255: *
256: CALL ZLARFT( 'Forward', 'Columnwise', M-I+1, IB,
257: $ A( I, I ), LDA, TAU( I ), WORK, LDWORK )
258: *
259: * Apply H**H to A(i:m,i+ib:n) from the left
260: *
261: CALL ZLARFB( 'Left', 'Conjugate transpose', 'Forward',
262: $ 'Columnwise', M-I+1, N-I-IB+1, IB,
263: $ A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
264: $ LDA, WORK( IB+1 ), LDWORK )
265: END IF
266: 10 CONTINUE
267: ELSE
268: I = 1
269: END IF
270: *
271: * Use unblocked code to factor the last or only block.
272: *
273: IF( I.LE.K )
274: $ CALL ZGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
275: $ IINFO )
276: *
277: WORK( 1 ) = IWS
278: RETURN
279: *
280: * End of ZGEQRF
281: *
282: END
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