1: SUBROUTINE ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
2: $ Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,
3: $ GIVCOL, GIVNUM, INFO )
4: *
5: * -- LAPACK routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
9: *
10: * .. Scalar Arguments ..
11: INTEGER CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ
12: DOUBLE PRECISION RHO
13: * ..
14: * .. Array Arguments ..
15: INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
16: $ INDXQ( * ), PERM( * )
17: DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
18: $ Z( * )
19: COMPLEX*16 Q( LDQ, * ), Q2( LDQ2, * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * ZLAED8 merges the two sets of eigenvalues together into a single
26: * sorted set. Then it tries to deflate the size of the problem.
27: * There are two ways in which deflation can occur: when two or more
28: * eigenvalues are close together or if there is a tiny element in the
29: * Z vector. For each such occurrence the order of the related secular
30: * equation problem is reduced by one.
31: *
32: * Arguments
33: * =========
34: *
35: * K (output) INTEGER
36: * Contains the number of non-deflated eigenvalues.
37: * This is the order of the related secular equation.
38: *
39: * N (input) INTEGER
40: * The dimension of the symmetric tridiagonal matrix. N >= 0.
41: *
42: * QSIZ (input) INTEGER
43: * The dimension of the unitary matrix used to reduce
44: * the dense or band matrix to tridiagonal form.
45: * QSIZ >= N if ICOMPQ = 1.
46: *
47: * Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
48: * On entry, Q contains the eigenvectors of the partially solved
49: * system which has been previously updated in matrix
50: * multiplies with other partially solved eigensystems.
51: * On exit, Q contains the trailing (N-K) updated eigenvectors
52: * (those which were deflated) in its last N-K columns.
53: *
54: * LDQ (input) INTEGER
55: * The leading dimension of the array Q. LDQ >= max( 1, N ).
56: *
57: * D (input/output) DOUBLE PRECISION array, dimension (N)
58: * On entry, D contains the eigenvalues of the two submatrices to
59: * be combined. On exit, D contains the trailing (N-K) updated
60: * eigenvalues (those which were deflated) sorted into increasing
61: * order.
62: *
63: * RHO (input/output) DOUBLE PRECISION
64: * Contains the off diagonal element associated with the rank-1
65: * cut which originally split the two submatrices which are now
66: * being recombined. RHO is modified during the computation to
67: * the value required by DLAED3.
68: *
69: * CUTPNT (input) INTEGER
70: * Contains the location of the last eigenvalue in the leading
71: * sub-matrix. MIN(1,N) <= CUTPNT <= N.
72: *
73: * Z (input) DOUBLE PRECISION array, dimension (N)
74: * On input this vector contains the updating vector (the last
75: * row of the first sub-eigenvector matrix and the first row of
76: * the second sub-eigenvector matrix). The contents of Z are
77: * destroyed during the updating process.
78: *
79: * DLAMDA (output) DOUBLE PRECISION array, dimension (N)
80: * Contains a copy of the first K eigenvalues which will be used
81: * by DLAED3 to form the secular equation.
82: *
83: * Q2 (output) COMPLEX*16 array, dimension (LDQ2,N)
84: * If ICOMPQ = 0, Q2 is not referenced. Otherwise,
85: * Contains a copy of the first K eigenvectors which will be used
86: * by DLAED7 in a matrix multiply (DGEMM) to update the new
87: * eigenvectors.
88: *
89: * LDQ2 (input) INTEGER
90: * The leading dimension of the array Q2. LDQ2 >= max( 1, N ).
91: *
92: * W (output) DOUBLE PRECISION array, dimension (N)
93: * This will hold the first k values of the final
94: * deflation-altered z-vector and will be passed to DLAED3.
95: *
96: * INDXP (workspace) INTEGER array, dimension (N)
97: * This will contain the permutation used to place deflated
98: * values of D at the end of the array. On output INDXP(1:K)
99: * points to the nondeflated D-values and INDXP(K+1:N)
100: * points to the deflated eigenvalues.
101: *
102: * INDX (workspace) INTEGER array, dimension (N)
103: * This will contain the permutation used to sort the contents of
104: * D into ascending order.
105: *
106: * INDXQ (input) INTEGER array, dimension (N)
107: * This contains the permutation which separately sorts the two
108: * sub-problems in D into ascending order. Note that elements in
109: * the second half of this permutation must first have CUTPNT
110: * added to their values in order to be accurate.
111: *
112: * PERM (output) INTEGER array, dimension (N)
113: * Contains the permutations (from deflation and sorting) to be
114: * applied to each eigenblock.
115: *
116: * GIVPTR (output) INTEGER
117: * Contains the number of Givens rotations which took place in
118: * this subproblem.
119: *
120: * GIVCOL (output) INTEGER array, dimension (2, N)
121: * Each pair of numbers indicates a pair of columns to take place
122: * in a Givens rotation.
123: *
124: * GIVNUM (output) DOUBLE PRECISION array, dimension (2, N)
125: * Each number indicates the S value to be used in the
126: * corresponding Givens rotation.
127: *
128: * INFO (output) INTEGER
129: * = 0: successful exit.
130: * < 0: if INFO = -i, the i-th argument had an illegal value.
131: *
132: * =====================================================================
133: *
134: * .. Parameters ..
135: DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
136: PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
137: $ TWO = 2.0D0, EIGHT = 8.0D0 )
138: * ..
139: * .. Local Scalars ..
140: INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
141: DOUBLE PRECISION C, EPS, S, T, TAU, TOL
142: * ..
143: * .. External Functions ..
144: INTEGER IDAMAX
145: DOUBLE PRECISION DLAMCH, DLAPY2
146: EXTERNAL IDAMAX, DLAMCH, DLAPY2
147: * ..
148: * .. External Subroutines ..
149: EXTERNAL DCOPY, DLAMRG, DSCAL, XERBLA, ZCOPY, ZDROT,
150: $ ZLACPY
151: * ..
152: * .. Intrinsic Functions ..
153: INTRINSIC ABS, MAX, MIN, SQRT
154: * ..
155: * .. Executable Statements ..
156: *
157: * Test the input parameters.
158: *
159: INFO = 0
160: *
161: IF( N.LT.0 ) THEN
162: INFO = -2
163: ELSE IF( QSIZ.LT.N ) THEN
164: INFO = -3
165: ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
166: INFO = -5
167: ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN
168: INFO = -8
169: ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN
170: INFO = -12
171: END IF
172: IF( INFO.NE.0 ) THEN
173: CALL XERBLA( 'ZLAED8', -INFO )
174: RETURN
175: END IF
176: *
177: * Quick return if possible
178: *
179: IF( N.EQ.0 )
180: $ RETURN
181: *
182: N1 = CUTPNT
183: N2 = N - N1
184: N1P1 = N1 + 1
185: *
186: IF( RHO.LT.ZERO ) THEN
187: CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
188: END IF
189: *
190: * Normalize z so that norm(z) = 1
191: *
192: T = ONE / SQRT( TWO )
193: DO 10 J = 1, N
194: INDX( J ) = J
195: 10 CONTINUE
196: CALL DSCAL( N, T, Z, 1 )
197: RHO = ABS( TWO*RHO )
198: *
199: * Sort the eigenvalues into increasing order
200: *
201: DO 20 I = CUTPNT + 1, N
202: INDXQ( I ) = INDXQ( I ) + CUTPNT
203: 20 CONTINUE
204: DO 30 I = 1, N
205: DLAMDA( I ) = D( INDXQ( I ) )
206: W( I ) = Z( INDXQ( I ) )
207: 30 CONTINUE
208: I = 1
209: J = CUTPNT + 1
210: CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDX )
211: DO 40 I = 1, N
212: D( I ) = DLAMDA( INDX( I ) )
213: Z( I ) = W( INDX( I ) )
214: 40 CONTINUE
215: *
216: * Calculate the allowable deflation tolerance
217: *
218: IMAX = IDAMAX( N, Z, 1 )
219: JMAX = IDAMAX( N, D, 1 )
220: EPS = DLAMCH( 'Epsilon' )
221: TOL = EIGHT*EPS*ABS( D( JMAX ) )
222: *
223: * If the rank-1 modifier is small enough, no more needs to be done
224: * -- except to reorganize Q so that its columns correspond with the
225: * elements in D.
226: *
227: IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
228: K = 0
229: DO 50 J = 1, N
230: PERM( J ) = INDXQ( INDX( J ) )
231: CALL ZCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
232: 50 CONTINUE
233: CALL ZLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ), LDQ )
234: RETURN
235: END IF
236: *
237: * If there are multiple eigenvalues then the problem deflates. Here
238: * the number of equal eigenvalues are found. As each equal
239: * eigenvalue is found, an elementary reflector is computed to rotate
240: * the corresponding eigensubspace so that the corresponding
241: * components of Z are zero in this new basis.
242: *
243: K = 0
244: GIVPTR = 0
245: K2 = N + 1
246: DO 60 J = 1, N
247: IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
248: *
249: * Deflate due to small z component.
250: *
251: K2 = K2 - 1
252: INDXP( K2 ) = J
253: IF( J.EQ.N )
254: $ GO TO 100
255: ELSE
256: JLAM = J
257: GO TO 70
258: END IF
259: 60 CONTINUE
260: 70 CONTINUE
261: J = J + 1
262: IF( J.GT.N )
263: $ GO TO 90
264: IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
265: *
266: * Deflate due to small z component.
267: *
268: K2 = K2 - 1
269: INDXP( K2 ) = J
270: ELSE
271: *
272: * Check if eigenvalues are close enough to allow deflation.
273: *
274: S = Z( JLAM )
275: C = Z( J )
276: *
277: * Find sqrt(a**2+b**2) without overflow or
278: * destructive underflow.
279: *
280: TAU = DLAPY2( C, S )
281: T = D( J ) - D( JLAM )
282: C = C / TAU
283: S = -S / TAU
284: IF( ABS( T*C*S ).LE.TOL ) THEN
285: *
286: * Deflation is possible.
287: *
288: Z( J ) = TAU
289: Z( JLAM ) = ZERO
290: *
291: * Record the appropriate Givens rotation
292: *
293: GIVPTR = GIVPTR + 1
294: GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) )
295: GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) )
296: GIVNUM( 1, GIVPTR ) = C
297: GIVNUM( 2, GIVPTR ) = S
298: CALL ZDROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
299: $ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
300: T = D( JLAM )*C*C + D( J )*S*S
301: D( J ) = D( JLAM )*S*S + D( J )*C*C
302: D( JLAM ) = T
303: K2 = K2 - 1
304: I = 1
305: 80 CONTINUE
306: IF( K2+I.LE.N ) THEN
307: IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN
308: INDXP( K2+I-1 ) = INDXP( K2+I )
309: INDXP( K2+I ) = JLAM
310: I = I + 1
311: GO TO 80
312: ELSE
313: INDXP( K2+I-1 ) = JLAM
314: END IF
315: ELSE
316: INDXP( K2+I-1 ) = JLAM
317: END IF
318: JLAM = J
319: ELSE
320: K = K + 1
321: W( K ) = Z( JLAM )
322: DLAMDA( K ) = D( JLAM )
323: INDXP( K ) = JLAM
324: JLAM = J
325: END IF
326: END IF
327: GO TO 70
328: 90 CONTINUE
329: *
330: * Record the last eigenvalue.
331: *
332: K = K + 1
333: W( K ) = Z( JLAM )
334: DLAMDA( K ) = D( JLAM )
335: INDXP( K ) = JLAM
336: *
337: 100 CONTINUE
338: *
339: * Sort the eigenvalues and corresponding eigenvectors into DLAMDA
340: * and Q2 respectively. The eigenvalues/vectors which were not
341: * deflated go into the first K slots of DLAMDA and Q2 respectively,
342: * while those which were deflated go into the last N - K slots.
343: *
344: DO 110 J = 1, N
345: JP = INDXP( J )
346: DLAMDA( J ) = D( JP )
347: PERM( J ) = INDXQ( INDX( JP ) )
348: CALL ZCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
349: 110 CONTINUE
350: *
351: * The deflated eigenvalues and their corresponding vectors go back
352: * into the last N - K slots of D and Q respectively.
353: *
354: IF( K.LT.N ) THEN
355: CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
356: CALL ZLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2, Q( 1, K+1 ),
357: $ LDQ )
358: END IF
359: *
360: RETURN
361: *
362: * End of ZLAED8
363: *
364: END
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