LAPACK  3.10.0 LAPACK: Linear Algebra PACKage
dlaed3.f
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1 *> \brief \b DLAED3 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
22 * CTOT, W, S, INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER INFO, K, LDQ, N, N1
26 * DOUBLE PRECISION RHO
27 * ..
28 * .. Array Arguments ..
29 * INTEGER CTOT( * ), INDX( * )
30 * DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
31 * \$ S( * ), W( * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> DLAED3 finds the roots of the secular equation, as defined by the
41 *> values in D, W, and RHO, between 1 and K. It makes the
42 *> appropriate calls to DLAED4 and then updates the eigenvectors by
43 *> multiplying the matrix of eigenvectors of the pair of eigensystems
44 *> being combined by the matrix of eigenvectors of the K-by-K system
45 *> which is solved here.
46 *>
47 *> This code makes very mild assumptions about floating point
48 *> arithmetic. It will work on machines with a guard digit in
49 *> add/subtract, or on those binary machines without guard digits
50 *> which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
51 *> It could conceivably fail on hexadecimal or decimal machines
52 *> without guard digits, but we know of none.
53 *> \endverbatim
54 *
55 * Arguments:
56 * ==========
57 *
58 *> \param[in] K
59 *> \verbatim
60 *> K is INTEGER
61 *> The number of terms in the rational function to be solved by
62 *> DLAED4. K >= 0.
63 *> \endverbatim
64 *>
65 *> \param[in] N
66 *> \verbatim
67 *> N is INTEGER
68 *> The number of rows and columns in the Q matrix.
69 *> N >= K (deflation may result in N>K).
70 *> \endverbatim
71 *>
72 *> \param[in] N1
73 *> \verbatim
74 *> N1 is INTEGER
75 *> The location of the last eigenvalue in the leading submatrix.
76 *> min(1,N) <= N1 <= N/2.
77 *> \endverbatim
78 *>
79 *> \param[out] D
80 *> \verbatim
81 *> D is DOUBLE PRECISION array, dimension (N)
82 *> D(I) contains the updated eigenvalues for
83 *> 1 <= I <= K.
84 *> \endverbatim
85 *>
86 *> \param[out] Q
87 *> \verbatim
88 *> Q is DOUBLE PRECISION array, dimension (LDQ,N)
89 *> Initially the first K columns are used as workspace.
90 *> On output the columns 1 to K contain
91 *> the updated eigenvectors.
92 *> \endverbatim
93 *>
94 *> \param[in] LDQ
95 *> \verbatim
96 *> LDQ is INTEGER
97 *> The leading dimension of the array Q. LDQ >= max(1,N).
98 *> \endverbatim
99 *>
100 *> \param[in] RHO
101 *> \verbatim
102 *> RHO is DOUBLE PRECISION
103 *> The value of the parameter in the rank one update equation.
104 *> RHO >= 0 required.
105 *> \endverbatim
106 *>
107 *> \param[in,out] DLAMDA
108 *> \verbatim
109 *> DLAMDA is DOUBLE PRECISION array, dimension (K)
110 *> The first K elements of this array contain the old roots
111 *> of the deflated updating problem. These are the poles
112 *> of the secular equation. May be changed on output by
113 *> having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
114 *> Cray-2, or Cray C-90, as described above.
115 *> \endverbatim
116 *>
117 *> \param[in] Q2
118 *> \verbatim
119 *> Q2 is DOUBLE PRECISION array, dimension (LDQ2*N)
120 *> The first K columns of this matrix contain the non-deflated
121 *> eigenvectors for the split problem.
122 *> \endverbatim
123 *>
124 *> \param[in] INDX
125 *> \verbatim
126 *> INDX is INTEGER array, dimension (N)
127 *> The permutation used to arrange the columns of the deflated
128 *> Q matrix into three groups (see DLAED2).
129 *> The rows of the eigenvectors found by DLAED4 must be likewise
130 *> permuted before the matrix multiply can take place.
131 *> \endverbatim
132 *>
133 *> \param[in] CTOT
134 *> \verbatim
135 *> CTOT is INTEGER array, dimension (4)
136 *> A count of the total number of the various types of columns
137 *> in Q, as described in INDX. The fourth column type is any
138 *> column which has been deflated.
139 *> \endverbatim
140 *>
141 *> \param[in,out] W
142 *> \verbatim
143 *> W is DOUBLE PRECISION array, dimension (K)
144 *> The first K elements of this array contain the components
145 *> of the deflation-adjusted updating vector. Destroyed on
146 *> output.
147 *> \endverbatim
148 *>
149 *> \param[out] S
150 *> \verbatim
151 *> S is DOUBLE PRECISION array, dimension (N1 + 1)*K
152 *> Will contain the eigenvectors of the repaired matrix which
153 *> will be multiplied by the previously accumulated eigenvectors
154 *> to update the system.
155 *> \endverbatim
156 *>
157 *> \param[out] INFO
158 *> \verbatim
159 *> INFO is INTEGER
160 *> = 0: successful exit.
161 *> < 0: if INFO = -i, the i-th argument had an illegal value.
162 *> > 0: if INFO = 1, an eigenvalue did not converge
163 *> \endverbatim
164 *
165 * Authors:
166 * ========
167 *
168 *> \author Univ. of Tennessee
169 *> \author Univ. of California Berkeley
170 *> \author Univ. of Colorado Denver
171 *> \author NAG Ltd.
172 *
173 *> \ingroup auxOTHERcomputational
174 *
175 *> \par Contributors:
176 * ==================
177 *>
178 *> Jeff Rutter, Computer Science Division, University of California
179 *> at Berkeley, USA \n
180 *> Modified by Francoise Tisseur, University of Tennessee
181 *>
182 * =====================================================================
183  SUBROUTINE dlaed3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
184  \$ CTOT, W, S, INFO )
185 *
186 * -- LAPACK computational routine --
187 * -- LAPACK is a software package provided by Univ. of Tennessee, --
188 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189 *
190 * .. Scalar Arguments ..
191  INTEGER INFO, K, LDQ, N, N1
192  DOUBLE PRECISION RHO
193 * ..
194 * .. Array Arguments ..
195  INTEGER CTOT( * ), INDX( * )
196  DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
197  \$ s( * ), w( * )
198 * ..
199 *
200 * =====================================================================
201 *
202 * .. Parameters ..
203  DOUBLE PRECISION ONE, ZERO
204  parameter( one = 1.0d0, zero = 0.0d0 )
205 * ..
206 * .. Local Scalars ..
207  INTEGER I, II, IQ2, J, N12, N2, N23
208  DOUBLE PRECISION TEMP
209 * ..
210 * .. External Functions ..
211  DOUBLE PRECISION DLAMC3, DNRM2
212  EXTERNAL dlamc3, dnrm2
213 * ..
214 * .. External Subroutines ..
215  EXTERNAL dcopy, dgemm, dlacpy, dlaed4, dlaset, xerbla
216 * ..
217 * .. Intrinsic Functions ..
218  INTRINSIC max, sign, sqrt
219 * ..
220 * .. Executable Statements ..
221 *
222 * Test the input parameters.
223 *
224  info = 0
225 *
226  IF( k.LT.0 ) THEN
227  info = -1
228  ELSE IF( n.LT.k ) THEN
229  info = -2
230  ELSE IF( ldq.LT.max( 1, n ) ) THEN
231  info = -6
232  END IF
233  IF( info.NE.0 ) THEN
234  CALL xerbla( 'DLAED3', -info )
235  RETURN
236  END IF
237 *
238 * Quick return if possible
239 *
240  IF( k.EQ.0 )
241  \$ RETURN
242 *
243 * Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
244 * be computed with high relative accuracy (barring over/underflow).
245 * This is a problem on machines without a guard digit in
246 * add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
247 * The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
248 * which on any of these machines zeros out the bottommost
249 * bit of DLAMDA(I) if it is 1; this makes the subsequent
250 * subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
251 * occurs. On binary machines with a guard digit (almost all
252 * machines) it does not change DLAMDA(I) at all. On hexadecimal
253 * and decimal machines with a guard digit, it slightly
254 * changes the bottommost bits of DLAMDA(I). It does not account
255 * for hexadecimal or decimal machines without guard digits
256 * (we know of none). We use a subroutine call to compute
257 * 2*DLAMBDA(I) to prevent optimizing compilers from eliminating
258 * this code.
259 *
260  DO 10 i = 1, k
261  dlamda( i ) = dlamc3( dlamda( i ), dlamda( i ) ) - dlamda( i )
262  10 CONTINUE
263 *
264  DO 20 j = 1, k
265  CALL dlaed4( k, j, dlamda, w, q( 1, j ), rho, d( j ), info )
266 *
267 * If the zero finder fails, the computation is terminated.
268 *
269  IF( info.NE.0 )
270  \$ GO TO 120
271  20 CONTINUE
272 *
273  IF( k.EQ.1 )
274  \$ GO TO 110
275  IF( k.EQ.2 ) THEN
276  DO 30 j = 1, k
277  w( 1 ) = q( 1, j )
278  w( 2 ) = q( 2, j )
279  ii = indx( 1 )
280  q( 1, j ) = w( ii )
281  ii = indx( 2 )
282  q( 2, j ) = w( ii )
283  30 CONTINUE
284  GO TO 110
285  END IF
286 *
287 * Compute updated W.
288 *
289  CALL dcopy( k, w, 1, s, 1 )
290 *
291 * Initialize W(I) = Q(I,I)
292 *
293  CALL dcopy( k, q, ldq+1, w, 1 )
294  DO 60 j = 1, k
295  DO 40 i = 1, j - 1
296  w( i ) = w( i )*( q( i, j ) / ( dlamda( i )-dlamda( j ) ) )
297  40 CONTINUE
298  DO 50 i = j + 1, k
299  w( i ) = w( i )*( q( i, j ) / ( dlamda( i )-dlamda( j ) ) )
300  50 CONTINUE
301  60 CONTINUE
302  DO 70 i = 1, k
303  w( i ) = sign( sqrt( -w( i ) ), s( i ) )
304  70 CONTINUE
305 *
306 * Compute eigenvectors of the modified rank-1 modification.
307 *
308  DO 100 j = 1, k
309  DO 80 i = 1, k
310  s( i ) = w( i ) / q( i, j )
311  80 CONTINUE
312  temp = dnrm2( k, s, 1 )
313  DO 90 i = 1, k
314  ii = indx( i )
315  q( i, j ) = s( ii ) / temp
316  90 CONTINUE
317  100 CONTINUE
318 *
319 * Compute the updated eigenvectors.
320 *
321  110 CONTINUE
322 *
323  n2 = n - n1
324  n12 = ctot( 1 ) + ctot( 2 )
325  n23 = ctot( 2 ) + ctot( 3 )
326 *
327  CALL dlacpy( 'A', n23, k, q( ctot( 1 )+1, 1 ), ldq, s, n23 )
328  iq2 = n1*n12 + 1
329  IF( n23.NE.0 ) THEN
330  CALL dgemm( 'N', 'N', n2, k, n23, one, q2( iq2 ), n2, s, n23,
331  \$ zero, q( n1+1, 1 ), ldq )
332  ELSE
333  CALL dlaset( 'A', n2, k, zero, zero, q( n1+1, 1 ), ldq )
334  END IF
335 *
336  CALL dlacpy( 'A', n12, k, q, ldq, s, n12 )
337  IF( n12.NE.0 ) THEN
338  CALL dgemm( 'N', 'N', n1, k, n12, one, q2, n1, s, n12, zero, q,
339  \$ ldq )
340  ELSE
341  CALL dlaset( 'A', n1, k, zero, zero, q( 1, 1 ), ldq )
342  END IF
343 *
344 *
345  120 CONTINUE
346  RETURN
347 *
348 * End of DLAED3
349 *
350  END
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dlaed4(N, I, D, Z, DELTA, RHO, DLAM, INFO)
DLAED4 used by DSTEDC. Finds a single root of the secular equation.
Definition: dlaed4.f:145
subroutine dlaed3(K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, CTOT, W, S, INFO)
DLAED3 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors....
Definition: dlaed3.f:185
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187