1: *> \brief \b ZGEQLF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGEQLF + dependencies
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGEQLF( 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: *> ZGEQLF computes a QL factorization of a complex M-by-N matrix A:
37: *> A = Q * L.
38: *> \endverbatim
39: *
40: * Arguments:
41: * ==========
42: *
43: *> \param[in] M
44: *> \verbatim
45: *> M is INTEGER
46: *> The number of rows of the matrix A. M >= 0.
47: *> \endverbatim
48: *>
49: *> \param[in] N
50: *> \verbatim
51: *> N is INTEGER
52: *> The number of columns of the matrix A. N >= 0.
53: *> \endverbatim
54: *>
55: *> \param[in,out] A
56: *> \verbatim
57: *> A is COMPLEX*16 array, dimension (LDA,N)
58: *> On entry, the M-by-N matrix A.
59: *> On exit,
60: *> if m >= n, the lower triangle of the subarray
61: *> A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
62: *> if m <= n, the elements on and below the (n-m)-th
63: *> superdiagonal contain the M-by-N lower trapezoidal matrix L;
64: *> the remaining elements, with the array TAU, represent the
65: *> unitary matrix Q as a product of elementary reflectors
66: *> (see Further Details).
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[out] TAU
76: *> \verbatim
77: *> TAU is COMPLEX*16 array, dimension (min(M,N))
78: *> The scalar factors of the elementary reflectors (see Further
79: *> Details).
80: *> \endverbatim
81: *>
82: *> \param[out] WORK
83: *> \verbatim
84: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
85: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
86: *> \endverbatim
87: *>
88: *> \param[in] LWORK
89: *> \verbatim
90: *> LWORK is INTEGER
91: *> The dimension of the array WORK. LWORK >= max(1,N).
92: *> For optimum performance LWORK >= N*NB, where NB is
93: *> the optimal blocksize.
94: *>
95: *> If LWORK = -1, then a workspace query is assumed; the routine
96: *> only calculates the optimal size of the WORK array, returns
97: *> this value as the first entry of the WORK array, and no error
98: *> message related to LWORK is issued by XERBLA.
99: *> \endverbatim
100: *>
101: *> \param[out] INFO
102: *> \verbatim
103: *> INFO is INTEGER
104: *> = 0: successful exit
105: *> < 0: if INFO = -i, the i-th argument had an illegal value
106: *> \endverbatim
107: *
108: * Authors:
109: * ========
110: *
111: *> \author Univ. of Tennessee
112: *> \author Univ. of California Berkeley
113: *> \author Univ. of Colorado Denver
114: *> \author NAG Ltd.
115: *
116: *> \ingroup complex16GEcomputational
117: *
118: *> \par Further Details:
119: * =====================
120: *>
121: *> \verbatim
122: *>
123: *> The matrix Q is represented as a product of elementary reflectors
124: *>
125: *> Q = H(k) . . . H(2) H(1), where k = min(m,n).
126: *>
127: *> Each H(i) has the form
128: *>
129: *> H(i) = I - tau * v * v**H
130: *>
131: *> where tau is a complex scalar, and v is a complex vector with
132: *> v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
133: *> A(1:m-k+i-1,n-k+i), and tau in TAU(i).
134: *> \endverbatim
135: *>
136: * =====================================================================
137: SUBROUTINE ZGEQLF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
138: *
139: * -- LAPACK computational routine --
140: * -- LAPACK is a software package provided by Univ. of Tennessee, --
141: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
142: *
143: * .. Scalar Arguments ..
144: INTEGER INFO, LDA, LWORK, M, N
145: * ..
146: * .. Array Arguments ..
147: COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
148: * ..
149: *
150: * =====================================================================
151: *
152: * .. Local Scalars ..
153: LOGICAL LQUERY
154: INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
155: $ MU, NB, NBMIN, NU, NX
156: * ..
157: * .. External Subroutines ..
158: EXTERNAL XERBLA, ZGEQL2, ZLARFB, ZLARFT
159: * ..
160: * .. Intrinsic Functions ..
161: INTRINSIC MAX, MIN
162: * ..
163: * .. External Functions ..
164: INTEGER ILAENV
165: EXTERNAL ILAENV
166: * ..
167: * .. Executable Statements ..
168: *
169: * Test the input arguments
170: *
171: INFO = 0
172: LQUERY = ( LWORK.EQ.-1 )
173: IF( M.LT.0 ) THEN
174: INFO = -1
175: ELSE IF( N.LT.0 ) THEN
176: INFO = -2
177: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
178: INFO = -4
179: END IF
180: *
181: IF( INFO.EQ.0 ) THEN
182: K = MIN( M, N )
183: IF( K.EQ.0 ) THEN
184: LWKOPT = 1
185: ELSE
186: NB = ILAENV( 1, 'ZGEQLF', ' ', M, N, -1, -1 )
187: LWKOPT = N*NB
188: END IF
189: WORK( 1 ) = LWKOPT
190: *
191: IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
192: INFO = -7
193: END IF
194: END IF
195: *
196: IF( INFO.NE.0 ) THEN
197: CALL XERBLA( 'ZGEQLF', -INFO )
198: RETURN
199: ELSE IF( LQUERY ) THEN
200: RETURN
201: END IF
202: *
203: * Quick return if possible
204: *
205: IF( K.EQ.0 ) THEN
206: RETURN
207: END IF
208: *
209: NBMIN = 2
210: NX = 1
211: IWS = N
212: IF( NB.GT.1 .AND. NB.LT.K ) THEN
213: *
214: * Determine when to cross over from blocked to unblocked code.
215: *
216: NX = MAX( 0, ILAENV( 3, 'ZGEQLF', ' ', M, N, -1, -1 ) )
217: IF( NX.LT.K ) THEN
218: *
219: * Determine if workspace is large enough for blocked code.
220: *
221: LDWORK = N
222: IWS = LDWORK*NB
223: IF( LWORK.LT.IWS ) THEN
224: *
225: * Not enough workspace to use optimal NB: reduce NB and
226: * determine the minimum value of NB.
227: *
228: NB = LWORK / LDWORK
229: NBMIN = MAX( 2, ILAENV( 2, 'ZGEQLF', ' ', M, N, -1,
230: $ -1 ) )
231: END IF
232: END IF
233: END IF
234: *
235: IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
236: *
237: * Use blocked code initially.
238: * The last kk columns are handled by the block method.
239: *
240: KI = ( ( K-NX-1 ) / NB )*NB
241: KK = MIN( K, KI+NB )
242: *
243: DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
244: IB = MIN( K-I+1, NB )
245: *
246: * Compute the QL factorization of the current block
247: * A(1:m-k+i+ib-1,n-k+i:n-k+i+ib-1)
248: *
249: CALL ZGEQL2( M-K+I+IB-1, IB, A( 1, N-K+I ), LDA, TAU( I ),
250: $ WORK, IINFO )
251: IF( N-K+I.GT.1 ) THEN
252: *
253: * Form the triangular factor of the block reflector
254: * H = H(i+ib-1) . . . H(i+1) H(i)
255: *
256: CALL ZLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB,
257: $ A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK )
258: *
259: * Apply H**H to A(1:m-k+i+ib-1,1:n-k+i-1) from the left
260: *
261: CALL ZLARFB( 'Left', 'Conjugate transpose', 'Backward',
262: $ 'Columnwise', M-K+I+IB-1, N-K+I-1, IB,
263: $ A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA,
264: $ WORK( IB+1 ), LDWORK )
265: END IF
266: 10 CONTINUE
267: MU = M - K + I + NB - 1
268: NU = N - K + I + NB - 1
269: ELSE
270: MU = M
271: NU = N
272: END IF
273: *
274: * Use unblocked code to factor the last or only block
275: *
276: IF( MU.GT.0 .AND. NU.GT.0 )
277: $ CALL ZGEQL2( MU, NU, A, LDA, TAU, WORK, IINFO )
278: *
279: WORK( 1 ) = IWS
280: RETURN
281: *
282: * End of ZGEQLF
283: *
284: END
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