Annotation of rpl/lapack/lapack/dlaed3.f, revision 1.6
1.1 bertrand 1: SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
2: $ CTOT, W, S, INFO )
3: *
4: * -- LAPACK routine (version 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
8: *
9: * .. Scalar Arguments ..
10: INTEGER INFO, K, LDQ, N, N1
11: DOUBLE PRECISION RHO
12: * ..
13: * .. Array Arguments ..
14: INTEGER CTOT( * ), INDX( * )
15: DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
16: $ S( * ), W( * )
17: * ..
18: *
19: * Purpose
20: * =======
21: *
22: * DLAED3 finds the roots of the secular equation, as defined by the
23: * values in D, W, and RHO, between 1 and K. It makes the
24: * appropriate calls to DLAED4 and then updates the eigenvectors by
25: * multiplying the matrix of eigenvectors of the pair of eigensystems
26: * being combined by the matrix of eigenvectors of the K-by-K system
27: * which is solved here.
28: *
29: * This code makes very mild assumptions about floating point
30: * arithmetic. It will work on machines with a guard digit in
31: * add/subtract, or on those binary machines without guard digits
32: * which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
33: * It could conceivably fail on hexadecimal or decimal machines
34: * without guard digits, but we know of none.
35: *
36: * Arguments
37: * =========
38: *
39: * K (input) INTEGER
40: * The number of terms in the rational function to be solved by
41: * DLAED4. K >= 0.
42: *
43: * N (input) INTEGER
44: * The number of rows and columns in the Q matrix.
45: * N >= K (deflation may result in N>K).
46: *
47: * N1 (input) INTEGER
48: * The location of the last eigenvalue in the leading submatrix.
49: * min(1,N) <= N1 <= N/2.
50: *
51: * D (output) DOUBLE PRECISION array, dimension (N)
52: * D(I) contains the updated eigenvalues for
53: * 1 <= I <= K.
54: *
55: * Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
56: * Initially the first K columns are used as workspace.
57: * On output the columns 1 to K contain
58: * the updated eigenvectors.
59: *
60: * LDQ (input) INTEGER
61: * The leading dimension of the array Q. LDQ >= max(1,N).
62: *
63: * RHO (input) DOUBLE PRECISION
64: * The value of the parameter in the rank one update equation.
65: * RHO >= 0 required.
66: *
67: * DLAMDA (input/output) DOUBLE PRECISION array, dimension (K)
68: * The first K elements of this array contain the old roots
69: * of the deflated updating problem. These are the poles
70: * of the secular equation. May be changed on output by
71: * having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
72: * Cray-2, or Cray C-90, as described above.
73: *
74: * Q2 (input) DOUBLE PRECISION array, dimension (LDQ2, N)
75: * The first K columns of this matrix contain the non-deflated
76: * eigenvectors for the split problem.
77: *
78: * INDX (input) INTEGER array, dimension (N)
79: * The permutation used to arrange the columns of the deflated
80: * Q matrix into three groups (see DLAED2).
81: * The rows of the eigenvectors found by DLAED4 must be likewise
82: * permuted before the matrix multiply can take place.
83: *
84: * CTOT (input) INTEGER array, dimension (4)
85: * A count of the total number of the various types of columns
86: * in Q, as described in INDX. The fourth column type is any
87: * column which has been deflated.
88: *
89: * W (input/output) DOUBLE PRECISION array, dimension (K)
90: * The first K elements of this array contain the components
91: * of the deflation-adjusted updating vector. Destroyed on
92: * output.
93: *
94: * S (workspace) DOUBLE PRECISION array, dimension (N1 + 1)*K
95: * Will contain the eigenvectors of the repaired matrix which
96: * will be multiplied by the previously accumulated eigenvectors
97: * to update the system.
98: *
99: * LDS (input) INTEGER
100: * The leading dimension of S. LDS >= max(1,K).
101: *
102: * INFO (output) INTEGER
103: * = 0: successful exit.
104: * < 0: if INFO = -i, the i-th argument had an illegal value.
105: * > 0: if INFO = 1, an eigenvalue did not converge
106: *
107: * Further Details
108: * ===============
109: *
110: * Based on contributions by
111: * Jeff Rutter, Computer Science Division, University of California
112: * at Berkeley, USA
113: * Modified by Francoise Tisseur, University of Tennessee.
114: *
115: * =====================================================================
116: *
117: * .. Parameters ..
118: DOUBLE PRECISION ONE, ZERO
119: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
120: * ..
121: * .. Local Scalars ..
122: INTEGER I, II, IQ2, J, N12, N2, N23
123: DOUBLE PRECISION TEMP
124: * ..
125: * .. External Functions ..
126: DOUBLE PRECISION DLAMC3, DNRM2
127: EXTERNAL DLAMC3, DNRM2
128: * ..
129: * .. External Subroutines ..
130: EXTERNAL DCOPY, DGEMM, DLACPY, DLAED4, DLASET, XERBLA
131: * ..
132: * .. Intrinsic Functions ..
133: INTRINSIC MAX, SIGN, SQRT
134: * ..
135: * .. Executable Statements ..
136: *
137: * Test the input parameters.
138: *
139: INFO = 0
140: *
141: IF( K.LT.0 ) THEN
142: INFO = -1
143: ELSE IF( N.LT.K ) THEN
144: INFO = -2
145: ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
146: INFO = -6
147: END IF
148: IF( INFO.NE.0 ) THEN
149: CALL XERBLA( 'DLAED3', -INFO )
150: RETURN
151: END IF
152: *
153: * Quick return if possible
154: *
155: IF( K.EQ.0 )
156: $ RETURN
157: *
158: * Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
159: * be computed with high relative accuracy (barring over/underflow).
160: * This is a problem on machines without a guard digit in
161: * add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
162: * The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
163: * which on any of these machines zeros out the bottommost
164: * bit of DLAMDA(I) if it is 1; this makes the subsequent
165: * subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
166: * occurs. On binary machines with a guard digit (almost all
167: * machines) it does not change DLAMDA(I) at all. On hexadecimal
168: * and decimal machines with a guard digit, it slightly
169: * changes the bottommost bits of DLAMDA(I). It does not account
170: * for hexadecimal or decimal machines without guard digits
171: * (we know of none). We use a subroutine call to compute
172: * 2*DLAMBDA(I) to prevent optimizing compilers from eliminating
173: * this code.
174: *
175: DO 10 I = 1, K
176: DLAMDA( I ) = DLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I )
177: 10 CONTINUE
178: *
179: DO 20 J = 1, K
180: CALL DLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO )
181: *
182: * If the zero finder fails, the computation is terminated.
183: *
184: IF( INFO.NE.0 )
185: $ GO TO 120
186: 20 CONTINUE
187: *
188: IF( K.EQ.1 )
189: $ GO TO 110
190: IF( K.EQ.2 ) THEN
191: DO 30 J = 1, K
192: W( 1 ) = Q( 1, J )
193: W( 2 ) = Q( 2, J )
194: II = INDX( 1 )
195: Q( 1, J ) = W( II )
196: II = INDX( 2 )
197: Q( 2, J ) = W( II )
198: 30 CONTINUE
199: GO TO 110
200: END IF
201: *
202: * Compute updated W.
203: *
204: CALL DCOPY( K, W, 1, S, 1 )
205: *
206: * Initialize W(I) = Q(I,I)
207: *
208: CALL DCOPY( K, Q, LDQ+1, W, 1 )
209: DO 60 J = 1, K
210: DO 40 I = 1, J - 1
211: W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
212: 40 CONTINUE
213: DO 50 I = J + 1, K
214: W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
215: 50 CONTINUE
216: 60 CONTINUE
217: DO 70 I = 1, K
218: W( I ) = SIGN( SQRT( -W( I ) ), S( I ) )
219: 70 CONTINUE
220: *
221: * Compute eigenvectors of the modified rank-1 modification.
222: *
223: DO 100 J = 1, K
224: DO 80 I = 1, K
225: S( I ) = W( I ) / Q( I, J )
226: 80 CONTINUE
227: TEMP = DNRM2( K, S, 1 )
228: DO 90 I = 1, K
229: II = INDX( I )
230: Q( I, J ) = S( II ) / TEMP
231: 90 CONTINUE
232: 100 CONTINUE
233: *
234: * Compute the updated eigenvectors.
235: *
236: 110 CONTINUE
237: *
238: N2 = N - N1
239: N12 = CTOT( 1 ) + CTOT( 2 )
240: N23 = CTOT( 2 ) + CTOT( 3 )
241: *
242: CALL DLACPY( 'A', N23, K, Q( CTOT( 1 )+1, 1 ), LDQ, S, N23 )
243: IQ2 = N1*N12 + 1
244: IF( N23.NE.0 ) THEN
245: CALL DGEMM( 'N', 'N', N2, K, N23, ONE, Q2( IQ2 ), N2, S, N23,
246: $ ZERO, Q( N1+1, 1 ), LDQ )
247: ELSE
248: CALL DLASET( 'A', N2, K, ZERO, ZERO, Q( N1+1, 1 ), LDQ )
249: END IF
250: *
251: CALL DLACPY( 'A', N12, K, Q, LDQ, S, N12 )
252: IF( N12.NE.0 ) THEN
253: CALL DGEMM( 'N', 'N', N1, K, N12, ONE, Q2, N1, S, N12, ZERO, Q,
254: $ LDQ )
255: ELSE
256: CALL DLASET( 'A', N1, K, ZERO, ZERO, Q( 1, 1 ), LDQ )
257: END IF
258: *
259: *
260: 120 CONTINUE
261: RETURN
262: *
263: * End of DLAED3
264: *
265: END
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