LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
dgehrd.f
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1 *> \brief \b DGEHRD
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER IHI, ILO, INFO, LDA, LWORK, N
25 * ..
26 * .. Array Arguments ..
27 * DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
28 * ..
29 *
30 *
31 *> \par Purpose:
32 * =============
33 *>
34 *> \verbatim
35 *>
36 *> DGEHRD reduces a real general matrix A to upper Hessenberg form H by
37 *> an orthogonal similarity transformation: Q**T * A * Q = H .
38 *> \endverbatim
39 *
40 * Arguments:
41 * ==========
42 *
43 *> \param[in] N
44 *> \verbatim
45 *> N is INTEGER
46 *> The order of the matrix A. N >= 0.
47 *> \endverbatim
48 *>
49 *> \param[in] ILO
50 *> \verbatim
51 *> ILO is INTEGER
52 *> \endverbatim
53 *>
54 *> \param[in] IHI
55 *> \verbatim
56 *> IHI is INTEGER
57 *>
58 *> It is assumed that A is already upper triangular in rows
59 *> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
60 *> set by a previous call to DGEBAL; otherwise they should be
61 *> set to 1 and N respectively. See Further Details.
62 *> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
63 *> \endverbatim
64 *>
65 *> \param[in,out] A
66 *> \verbatim
67 *> A is DOUBLE PRECISION array, dimension (LDA,N)
68 *> On entry, the N-by-N general matrix to be reduced.
69 *> On exit, the upper triangle and the first subdiagonal of A
70 *> are overwritten with the upper Hessenberg matrix H, and the
71 *> elements below the first subdiagonal, with the array TAU,
72 *> represent the orthogonal matrix Q as a product of elementary
73 *> reflectors. See Further 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,N).
80 *> \endverbatim
81 *>
82 *> \param[out] TAU
83 *> \verbatim
84 *> TAU is DOUBLE PRECISION array, dimension (N-1)
85 *> The scalar factors of the elementary reflectors (see Further
86 *> Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
87 *> zero.
88 *> \endverbatim
89 *>
90 *> \param[out] WORK
91 *> \verbatim
92 *> WORK is DOUBLE PRECISION array, dimension (LWORK)
93 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
94 *> \endverbatim
95 *>
96 *> \param[in] LWORK
97 *> \verbatim
98 *> LWORK is INTEGER
99 *> The length of the array WORK. LWORK >= max(1,N).
100 *> For good performance, LWORK should generally be larger.
101 *>
102 *> If LWORK = -1, then a workspace query is assumed; the routine
103 *> only calculates the optimal size of the WORK array, returns
104 *> this value as the first entry of the WORK array, and no error
105 *> message related to LWORK is issued by XERBLA.
106 *> \endverbatim
107 *>
108 *> \param[out] INFO
109 *> \verbatim
110 *> INFO is INTEGER
111 *> = 0: successful exit
112 *> < 0: if INFO = -i, the i-th argument had an illegal value.
113 *> \endverbatim
114 *
115 * Authors:
116 * ========
117 *
118 *> \author Univ. of Tennessee
119 *> \author Univ. of California Berkeley
120 *> \author Univ. of Colorado Denver
121 *> \author NAG Ltd.
122 *
123 *> \ingroup doubleGEcomputational
124 *
125 *> \par Further Details:
126 * =====================
127 *>
128 *> \verbatim
129 *>
130 *> The matrix Q is represented as a product of (ihi-ilo) elementary
131 *> reflectors
132 *>
133 *> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
134 *>
135 *> Each H(i) has the form
136 *>
137 *> H(i) = I - tau * v * v**T
138 *>
139 *> where tau is a real scalar, and v is a real vector with
140 *> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
141 *> exit in A(i+2:ihi,i), and tau in TAU(i).
142 *>
143 *> The contents of A are illustrated by the following example, with
144 *> n = 7, ilo = 2 and ihi = 6:
145 *>
146 *> on entry, on exit,
147 *>
148 *> ( a a a a a a a ) ( a a h h h h a )
149 *> ( a a a a a a ) ( a h h h h a )
150 *> ( a a a a a a ) ( h h h h h h )
151 *> ( a a a a a a ) ( v2 h h h h h )
152 *> ( a a a a a a ) ( v2 v3 h h h h )
153 *> ( a a a a a a ) ( v2 v3 v4 h h h )
154 *> ( a ) ( a )
155 *>
156 *> where a denotes an element of the original matrix A, h denotes a
157 *> modified element of the upper Hessenberg matrix H, and vi denotes an
158 *> element of the vector defining H(i).
159 *>
160 *> This file is a slight modification of LAPACK-3.0's DGEHRD
161 *> subroutine incorporating improvements proposed by Quintana-Orti and
162 *> Van de Geijn (2006). (See DLAHR2.)
163 *> \endverbatim
164 *>
165 * =====================================================================
166  SUBROUTINE dgehrd( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
167 *
168 * -- LAPACK computational routine --
169 * -- LAPACK is a software package provided by Univ. of Tennessee, --
170 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
171 *
172 * .. Scalar Arguments ..
173  INTEGER IHI, ILO, INFO, LDA, LWORK, N
174 * ..
175 * .. Array Arguments ..
176  DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
177 * ..
178 *
179 * =====================================================================
180 *
181 * .. Parameters ..
182  INTEGER NBMAX, LDT, TSIZE
183  parameter( nbmax = 64, ldt = nbmax+1,
184  \$ tsize = ldt*nbmax )
185  DOUBLE PRECISION ZERO, ONE
186  parameter( zero = 0.0d+0,
187  \$ one = 1.0d+0 )
188 * ..
189 * .. Local Scalars ..
190  LOGICAL LQUERY
191  INTEGER I, IB, IINFO, IWT, J, LDWORK, LWKOPT, NB,
192  \$ NBMIN, NH, NX
193  DOUBLE PRECISION EI
194 * ..
195 * .. External Subroutines ..
196  EXTERNAL daxpy, dgehd2, dgemm, dlahr2, dlarfb, dtrmm,
197  \$ xerbla
198 * ..
199 * .. Intrinsic Functions ..
200  INTRINSIC max, min
201 * ..
202 * .. External Functions ..
203  INTEGER ILAENV
204  EXTERNAL ilaenv
205 * ..
206 * .. Executable Statements ..
207 *
208 * Test the input parameters
209 *
210  info = 0
211  lquery = ( lwork.EQ.-1 )
212  IF( n.LT.0 ) THEN
213  info = -1
214  ELSE IF( ilo.LT.1 .OR. ilo.GT.max( 1, n ) ) THEN
215  info = -2
216  ELSE IF( ihi.LT.min( ilo, n ) .OR. ihi.GT.n ) THEN
217  info = -3
218  ELSE IF( lda.LT.max( 1, n ) ) THEN
219  info = -5
220  ELSE IF( lwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
221  info = -8
222  END IF
223 *
224  IF( info.EQ.0 ) THEN
225 *
226 * Compute the workspace requirements
227 *
228  nb = min( nbmax, ilaenv( 1, 'DGEHRD', ' ', n, ilo, ihi, -1 ) )
229  lwkopt = n*nb + tsize
230  work( 1 ) = lwkopt
231  END IF
232 *
233  IF( info.NE.0 ) THEN
234  CALL xerbla( 'DGEHRD', -info )
235  RETURN
236  ELSE IF( lquery ) THEN
237  RETURN
238  END IF
239 *
240 * Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
241 *
242  DO 10 i = 1, ilo - 1
243  tau( i ) = zero
244  10 CONTINUE
245  DO 20 i = max( 1, ihi ), n - 1
246  tau( i ) = zero
247  20 CONTINUE
248 *
249 * Quick return if possible
250 *
251  nh = ihi - ilo + 1
252  IF( nh.LE.1 ) THEN
253  work( 1 ) = 1
254  RETURN
255  END IF
256 *
257 * Determine the block size
258 *
259  nb = min( nbmax, ilaenv( 1, 'DGEHRD', ' ', n, ilo, ihi, -1 ) )
260  nbmin = 2
261  IF( nb.GT.1 .AND. nb.LT.nh ) THEN
262 *
263 * Determine when to cross over from blocked to unblocked code
264 * (last block is always handled by unblocked code)
265 *
266  nx = max( nb, ilaenv( 3, 'DGEHRD', ' ', n, ilo, ihi, -1 ) )
267  IF( nx.LT.nh ) THEN
268 *
269 * Determine if workspace is large enough for blocked code
270 *
271  IF( lwork.LT.n*nb+tsize ) THEN
272 *
273 * Not enough workspace to use optimal NB: determine the
274 * minimum value of NB, and reduce NB or force use of
275 * unblocked code
276 *
277  nbmin = max( 2, ilaenv( 2, 'DGEHRD', ' ', n, ilo, ihi,
278  \$ -1 ) )
279  IF( lwork.GE.(n*nbmin + tsize) ) THEN
280  nb = (lwork-tsize) / n
281  ELSE
282  nb = 1
283  END IF
284  END IF
285  END IF
286  END IF
287  ldwork = n
288 *
289  IF( nb.LT.nbmin .OR. nb.GE.nh ) THEN
290 *
291 * Use unblocked code below
292 *
293  i = ilo
294 *
295  ELSE
296 *
297 * Use blocked code
298 *
299  iwt = 1 + n*nb
300  DO 40 i = ilo, ihi - 1 - nx, nb
301  ib = min( nb, ihi-i )
302 *
303 * Reduce columns i:i+ib-1 to Hessenberg form, returning the
304 * matrices V and T of the block reflector H = I - V*T*V**T
305 * which performs the reduction, and also the matrix Y = A*V*T
306 *
307  CALL dlahr2( ihi, i, ib, a( 1, i ), lda, tau( i ),
308  \$ work( iwt ), ldt, work, ldwork )
309 *
310 * Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
311 * right, computing A := A - Y * V**T. V(i+ib,ib-1) must be set
312 * to 1
313 *
314  ei = a( i+ib, i+ib-1 )
315  a( i+ib, i+ib-1 ) = one
316  CALL dgemm( 'No transpose', 'Transpose',
317  \$ ihi, ihi-i-ib+1,
318  \$ ib, -one, work, ldwork, a( i+ib, i ), lda, one,
319  \$ a( 1, i+ib ), lda )
320  a( i+ib, i+ib-1 ) = ei
321 *
322 * Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
323 * right
324 *
325  CALL dtrmm( 'Right', 'Lower', 'Transpose',
326  \$ 'Unit', i, ib-1,
327  \$ one, a( i+1, i ), lda, work, ldwork )
328  DO 30 j = 0, ib-2
329  CALL daxpy( i, -one, work( ldwork*j+1 ), 1,
330  \$ a( 1, i+j+1 ), 1 )
331  30 CONTINUE
332 *
333 * Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
334 * left
335 *
336  CALL dlarfb( 'Left', 'Transpose', 'Forward',
337  \$ 'Columnwise',
338  \$ ihi-i, n-i-ib+1, ib, a( i+1, i ), lda,
339  \$ work( iwt ), ldt, a( i+1, i+ib ), lda,
340  \$ work, ldwork )
341  40 CONTINUE
342  END IF
343 *
344 * Use unblocked code to reduce the rest of the matrix
345 *
346  CALL dgehd2( n, i, ihi, a, lda, tau, work, iinfo )
347  work( 1 ) = lwkopt
348 *
349  RETURN
350 *
351 * End of DGEHRD
352 *
353  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
subroutine dtrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRMM
Definition: dtrmm.f:177
subroutine dgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
DGEHRD
Definition: dgehrd.f:167
subroutine dgehd2(N, ILO, IHI, A, LDA, TAU, WORK, INFO)
DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
Definition: dgehd2.f:149
subroutine dlarfb(SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
DLARFB applies a block reflector or its transpose to a general rectangular matrix.
Definition: dlarfb.f:197
subroutine dlahr2(N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elemen...
Definition: dlahr2.f:181