1: *> \brief \b DLAED3
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLAED3 + 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/dlaed3.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
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: *> \date November 2011
174: *
175: *> \ingroup auxOTHERcomputational
176: *
177: *> \par Contributors:
178: * ==================
179: *>
180: *> Jeff Rutter, Computer Science Division, University of California
181: *> at Berkeley, USA \n
182: *> Modified by Francoise Tisseur, University of Tennessee
183: *>
184: * =====================================================================
185: SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
186: $ CTOT, W, S, INFO )
187: *
188: * -- LAPACK computational routine (version 3.4.0) --
189: * -- LAPACK is a software package provided by Univ. of Tennessee, --
190: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
191: * November 2011
192: *
193: * .. Scalar Arguments ..
194: INTEGER INFO, K, LDQ, N, N1
195: DOUBLE PRECISION RHO
196: * ..
197: * .. Array Arguments ..
198: INTEGER CTOT( * ), INDX( * )
199: DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
200: $ S( * ), W( * )
201: * ..
202: *
203: * =====================================================================
204: *
205: * .. Parameters ..
206: DOUBLE PRECISION ONE, ZERO
207: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
208: * ..
209: * .. Local Scalars ..
210: INTEGER I, II, IQ2, J, N12, N2, N23
211: DOUBLE PRECISION TEMP
212: * ..
213: * .. External Functions ..
214: DOUBLE PRECISION DLAMC3, DNRM2
215: EXTERNAL DLAMC3, DNRM2
216: * ..
217: * .. External Subroutines ..
218: EXTERNAL DCOPY, DGEMM, DLACPY, DLAED4, DLASET, XERBLA
219: * ..
220: * .. Intrinsic Functions ..
221: INTRINSIC MAX, SIGN, SQRT
222: * ..
223: * .. Executable Statements ..
224: *
225: * Test the input parameters.
226: *
227: INFO = 0
228: *
229: IF( K.LT.0 ) THEN
230: INFO = -1
231: ELSE IF( N.LT.K ) THEN
232: INFO = -2
233: ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
234: INFO = -6
235: END IF
236: IF( INFO.NE.0 ) THEN
237: CALL XERBLA( 'DLAED3', -INFO )
238: RETURN
239: END IF
240: *
241: * Quick return if possible
242: *
243: IF( K.EQ.0 )
244: $ RETURN
245: *
246: * Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
247: * be computed with high relative accuracy (barring over/underflow).
248: * This is a problem on machines without a guard digit in
249: * add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
250: * The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
251: * which on any of these machines zeros out the bottommost
252: * bit of DLAMDA(I) if it is 1; this makes the subsequent
253: * subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
254: * occurs. On binary machines with a guard digit (almost all
255: * machines) it does not change DLAMDA(I) at all. On hexadecimal
256: * and decimal machines with a guard digit, it slightly
257: * changes the bottommost bits of DLAMDA(I). It does not account
258: * for hexadecimal or decimal machines without guard digits
259: * (we know of none). We use a subroutine call to compute
260: * 2*DLAMBDA(I) to prevent optimizing compilers from eliminating
261: * this code.
262: *
263: DO 10 I = 1, K
264: DLAMDA( I ) = DLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I )
265: 10 CONTINUE
266: *
267: DO 20 J = 1, K
268: CALL DLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO )
269: *
270: * If the zero finder fails, the computation is terminated.
271: *
272: IF( INFO.NE.0 )
273: $ GO TO 120
274: 20 CONTINUE
275: *
276: IF( K.EQ.1 )
277: $ GO TO 110
278: IF( K.EQ.2 ) THEN
279: DO 30 J = 1, K
280: W( 1 ) = Q( 1, J )
281: W( 2 ) = Q( 2, J )
282: II = INDX( 1 )
283: Q( 1, J ) = W( II )
284: II = INDX( 2 )
285: Q( 2, J ) = W( II )
286: 30 CONTINUE
287: GO TO 110
288: END IF
289: *
290: * Compute updated W.
291: *
292: CALL DCOPY( K, W, 1, S, 1 )
293: *
294: * Initialize W(I) = Q(I,I)
295: *
296: CALL DCOPY( K, Q, LDQ+1, W, 1 )
297: DO 60 J = 1, K
298: DO 40 I = 1, J - 1
299: W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
300: 40 CONTINUE
301: DO 50 I = J + 1, K
302: W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
303: 50 CONTINUE
304: 60 CONTINUE
305: DO 70 I = 1, K
306: W( I ) = SIGN( SQRT( -W( I ) ), S( I ) )
307: 70 CONTINUE
308: *
309: * Compute eigenvectors of the modified rank-1 modification.
310: *
311: DO 100 J = 1, K
312: DO 80 I = 1, K
313: S( I ) = W( I ) / Q( I, J )
314: 80 CONTINUE
315: TEMP = DNRM2( K, S, 1 )
316: DO 90 I = 1, K
317: II = INDX( I )
318: Q( I, J ) = S( II ) / TEMP
319: 90 CONTINUE
320: 100 CONTINUE
321: *
322: * Compute the updated eigenvectors.
323: *
324: 110 CONTINUE
325: *
326: N2 = N - N1
327: N12 = CTOT( 1 ) + CTOT( 2 )
328: N23 = CTOT( 2 ) + CTOT( 3 )
329: *
330: CALL DLACPY( 'A', N23, K, Q( CTOT( 1 )+1, 1 ), LDQ, S, N23 )
331: IQ2 = N1*N12 + 1
332: IF( N23.NE.0 ) THEN
333: CALL DGEMM( 'N', 'N', N2, K, N23, ONE, Q2( IQ2 ), N2, S, N23,
334: $ ZERO, Q( N1+1, 1 ), LDQ )
335: ELSE
336: CALL DLASET( 'A', N2, K, ZERO, ZERO, Q( N1+1, 1 ), LDQ )
337: END IF
338: *
339: CALL DLACPY( 'A', N12, K, Q, LDQ, S, N12 )
340: IF( N12.NE.0 ) THEN
341: CALL DGEMM( 'N', 'N', N1, K, N12, ONE, Q2, N1, S, N12, ZERO, Q,
342: $ LDQ )
343: ELSE
344: CALL DLASET( 'A', N1, K, ZERO, ZERO, Q( 1, 1 ), LDQ )
345: END IF
346: *
347: *
348: 120 CONTINUE
349: RETURN
350: *
351: * End of DLAED3
352: *
353: END
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