1: SUBROUTINE ZSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
2: $ IWORK, IFAIL, 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, LDZ, M, N
11: * ..
12: * .. Array Arguments ..
13: INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
14: $ IWORK( * )
15: DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
16: COMPLEX*16 Z( LDZ, * )
17: * ..
18: *
19: * Purpose
20: * =======
21: *
22: * ZSTEIN computes the eigenvectors of a real symmetric tridiagonal
23: * matrix T corresponding to specified eigenvalues, using inverse
24: * iteration.
25: *
26: * The maximum number of iterations allowed for each eigenvector is
27: * specified by an internal parameter MAXITS (currently set to 5).
28: *
29: * Although the eigenvectors are real, they are stored in a complex
30: * array, which may be passed to ZUNMTR or ZUPMTR for back
31: * transformation to the eigenvectors of a complex Hermitian matrix
32: * which was reduced to tridiagonal form.
33: *
34: *
35: * Arguments
36: * =========
37: *
38: * N (input) INTEGER
39: * The order of the matrix. N >= 0.
40: *
41: * D (input) DOUBLE PRECISION array, dimension (N)
42: * The n diagonal elements of the tridiagonal matrix T.
43: *
44: * E (input) DOUBLE PRECISION array, dimension (N-1)
45: * The (n-1) subdiagonal elements of the tridiagonal matrix
46: * T, stored in elements 1 to N-1.
47: *
48: * M (input) INTEGER
49: * The number of eigenvectors to be found. 0 <= M <= N.
50: *
51: * W (input) DOUBLE PRECISION array, dimension (N)
52: * The first M elements of W contain the eigenvalues for
53: * which eigenvectors are to be computed. The eigenvalues
54: * should be grouped by split-off block and ordered from
55: * smallest to largest within the block. ( The output array
56: * W from DSTEBZ with ORDER = 'B' is expected here. )
57: *
58: * IBLOCK (input) INTEGER array, dimension (N)
59: * The submatrix indices associated with the corresponding
60: * eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
61: * the first submatrix from the top, =2 if W(i) belongs to
62: * the second submatrix, etc. ( The output array IBLOCK
63: * from DSTEBZ is expected here. )
64: *
65: * ISPLIT (input) INTEGER array, dimension (N)
66: * The splitting points, at which T breaks up into submatrices.
67: * The first submatrix consists of rows/columns 1 to
68: * ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
69: * through ISPLIT( 2 ), etc.
70: * ( The output array ISPLIT from DSTEBZ is expected here. )
71: *
72: * Z (output) COMPLEX*16 array, dimension (LDZ, M)
73: * The computed eigenvectors. The eigenvector associated
74: * with the eigenvalue W(i) is stored in the i-th column of
75: * Z. Any vector which fails to converge is set to its current
76: * iterate after MAXITS iterations.
77: * The imaginary parts of the eigenvectors are set to zero.
78: *
79: * LDZ (input) INTEGER
80: * The leading dimension of the array Z. LDZ >= max(1,N).
81: *
82: * WORK (workspace) DOUBLE PRECISION array, dimension (5*N)
83: *
84: * IWORK (workspace) INTEGER array, dimension (N)
85: *
86: * IFAIL (output) INTEGER array, dimension (M)
87: * On normal exit, all elements of IFAIL are zero.
88: * If one or more eigenvectors fail to converge after
89: * MAXITS iterations, then their indices are stored in
90: * array IFAIL.
91: *
92: * INFO (output) INTEGER
93: * = 0: successful exit
94: * < 0: if INFO = -i, the i-th argument had an illegal value
95: * > 0: if INFO = i, then i eigenvectors failed to converge
96: * in MAXITS iterations. Their indices are stored in
97: * array IFAIL.
98: *
99: * Internal Parameters
100: * ===================
101: *
102: * MAXITS INTEGER, default = 5
103: * The maximum number of iterations performed.
104: *
105: * EXTRA INTEGER, default = 2
106: * The number of iterations performed after norm growth
107: * criterion is satisfied, should be at least 1.
108: *
109: * =====================================================================
110: *
111: * .. Parameters ..
112: COMPLEX*16 CZERO, CONE
113: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
114: $ CONE = ( 1.0D+0, 0.0D+0 ) )
115: DOUBLE PRECISION ZERO, ONE, TEN, ODM3, ODM1
116: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 1.0D+1,
117: $ ODM3 = 1.0D-3, ODM1 = 1.0D-1 )
118: INTEGER MAXITS, EXTRA
119: PARAMETER ( MAXITS = 5, EXTRA = 2 )
120: * ..
121: * .. Local Scalars ..
122: INTEGER B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1,
123: $ INDRV2, INDRV3, INDRV4, INDRV5, ITS, J, J1,
124: $ JBLK, JMAX, JR, NBLK, NRMCHK
125: DOUBLE PRECISION DTPCRT, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL,
126: $ SCL, SEP, TOL, XJ, XJM, ZTR
127: * ..
128: * .. Local Arrays ..
129: INTEGER ISEED( 4 )
130: * ..
131: * .. External Functions ..
132: INTEGER IDAMAX
133: DOUBLE PRECISION DASUM, DLAMCH, DNRM2
134: EXTERNAL IDAMAX, DASUM, DLAMCH, DNRM2
135: * ..
136: * .. External Subroutines ..
137: EXTERNAL DCOPY, DLAGTF, DLAGTS, DLARNV, DSCAL, XERBLA
138: * ..
139: * .. Intrinsic Functions ..
140: INTRINSIC ABS, DBLE, DCMPLX, MAX, SQRT
141: * ..
142: * .. Executable Statements ..
143: *
144: * Test the input parameters.
145: *
146: INFO = 0
147: DO 10 I = 1, M
148: IFAIL( I ) = 0
149: 10 CONTINUE
150: *
151: IF( N.LT.0 ) THEN
152: INFO = -1
153: ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
154: INFO = -4
155: ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN
156: INFO = -9
157: ELSE
158: DO 20 J = 2, M
159: IF( IBLOCK( J ).LT.IBLOCK( J-1 ) ) THEN
160: INFO = -6
161: GO TO 30
162: END IF
163: IF( IBLOCK( J ).EQ.IBLOCK( J-1 ) .AND. W( J ).LT.W( J-1 ) )
164: $ THEN
165: INFO = -5
166: GO TO 30
167: END IF
168: 20 CONTINUE
169: 30 CONTINUE
170: END IF
171: *
172: IF( INFO.NE.0 ) THEN
173: CALL XERBLA( 'ZSTEIN', -INFO )
174: RETURN
175: END IF
176: *
177: * Quick return if possible
178: *
179: IF( N.EQ.0 .OR. M.EQ.0 ) THEN
180: RETURN
181: ELSE IF( N.EQ.1 ) THEN
182: Z( 1, 1 ) = CONE
183: RETURN
184: END IF
185: *
186: * Get machine constants.
187: *
188: EPS = DLAMCH( 'Precision' )
189: *
190: * Initialize seed for random number generator DLARNV.
191: *
192: DO 40 I = 1, 4
193: ISEED( I ) = 1
194: 40 CONTINUE
195: *
196: * Initialize pointers.
197: *
198: INDRV1 = 0
199: INDRV2 = INDRV1 + N
200: INDRV3 = INDRV2 + N
201: INDRV4 = INDRV3 + N
202: INDRV5 = INDRV4 + N
203: *
204: * Compute eigenvectors of matrix blocks.
205: *
206: J1 = 1
207: DO 180 NBLK = 1, IBLOCK( M )
208: *
209: * Find starting and ending indices of block nblk.
210: *
211: IF( NBLK.EQ.1 ) THEN
212: B1 = 1
213: ELSE
214: B1 = ISPLIT( NBLK-1 ) + 1
215: END IF
216: BN = ISPLIT( NBLK )
217: BLKSIZ = BN - B1 + 1
218: IF( BLKSIZ.EQ.1 )
219: $ GO TO 60
220: GPIND = B1
221: *
222: * Compute reorthogonalization criterion and stopping criterion.
223: *
224: ONENRM = ABS( D( B1 ) ) + ABS( E( B1 ) )
225: ONENRM = MAX( ONENRM, ABS( D( BN ) )+ABS( E( BN-1 ) ) )
226: DO 50 I = B1 + 1, BN - 1
227: ONENRM = MAX( ONENRM, ABS( D( I ) )+ABS( E( I-1 ) )+
228: $ ABS( E( I ) ) )
229: 50 CONTINUE
230: ORTOL = ODM3*ONENRM
231: *
232: DTPCRT = SQRT( ODM1 / BLKSIZ )
233: *
234: * Loop through eigenvalues of block nblk.
235: *
236: 60 CONTINUE
237: JBLK = 0
238: DO 170 J = J1, M
239: IF( IBLOCK( J ).NE.NBLK ) THEN
240: J1 = J
241: GO TO 180
242: END IF
243: JBLK = JBLK + 1
244: XJ = W( J )
245: *
246: * Skip all the work if the block size is one.
247: *
248: IF( BLKSIZ.EQ.1 ) THEN
249: WORK( INDRV1+1 ) = ONE
250: GO TO 140
251: END IF
252: *
253: * If eigenvalues j and j-1 are too close, add a relatively
254: * small perturbation.
255: *
256: IF( JBLK.GT.1 ) THEN
257: EPS1 = ABS( EPS*XJ )
258: PERTOL = TEN*EPS1
259: SEP = XJ - XJM
260: IF( SEP.LT.PERTOL )
261: $ XJ = XJM + PERTOL
262: END IF
263: *
264: ITS = 0
265: NRMCHK = 0
266: *
267: * Get random starting vector.
268: *
269: CALL DLARNV( 2, ISEED, BLKSIZ, WORK( INDRV1+1 ) )
270: *
271: * Copy the matrix T so it won't be destroyed in factorization.
272: *
273: CALL DCOPY( BLKSIZ, D( B1 ), 1, WORK( INDRV4+1 ), 1 )
274: CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV2+2 ), 1 )
275: CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV3+1 ), 1 )
276: *
277: * Compute LU factors with partial pivoting ( PT = LU )
278: *
279: TOL = ZERO
280: CALL DLAGTF( BLKSIZ, WORK( INDRV4+1 ), XJ, WORK( INDRV2+2 ),
281: $ WORK( INDRV3+1 ), TOL, WORK( INDRV5+1 ), IWORK,
282: $ IINFO )
283: *
284: * Update iteration count.
285: *
286: 70 CONTINUE
287: ITS = ITS + 1
288: IF( ITS.GT.MAXITS )
289: $ GO TO 120
290: *
291: * Normalize and scale the righthand side vector Pb.
292: *
293: SCL = BLKSIZ*ONENRM*MAX( EPS,
294: $ ABS( WORK( INDRV4+BLKSIZ ) ) ) /
295: $ DASUM( BLKSIZ, WORK( INDRV1+1 ), 1 )
296: CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
297: *
298: * Solve the system LU = Pb.
299: *
300: CALL DLAGTS( -1, BLKSIZ, WORK( INDRV4+1 ), WORK( INDRV2+2 ),
301: $ WORK( INDRV3+1 ), WORK( INDRV5+1 ), IWORK,
302: $ WORK( INDRV1+1 ), TOL, IINFO )
303: *
304: * Reorthogonalize by modified Gram-Schmidt if eigenvalues are
305: * close enough.
306: *
307: IF( JBLK.EQ.1 )
308: $ GO TO 110
309: IF( ABS( XJ-XJM ).GT.ORTOL )
310: $ GPIND = J
311: IF( GPIND.NE.J ) THEN
312: DO 100 I = GPIND, J - 1
313: ZTR = ZERO
314: DO 80 JR = 1, BLKSIZ
315: ZTR = ZTR + WORK( INDRV1+JR )*
316: $ DBLE( Z( B1-1+JR, I ) )
317: 80 CONTINUE
318: DO 90 JR = 1, BLKSIZ
319: WORK( INDRV1+JR ) = WORK( INDRV1+JR ) -
320: $ ZTR*DBLE( Z( B1-1+JR, I ) )
321: 90 CONTINUE
322: 100 CONTINUE
323: END IF
324: *
325: * Check the infinity norm of the iterate.
326: *
327: 110 CONTINUE
328: JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
329: NRM = ABS( WORK( INDRV1+JMAX ) )
330: *
331: * Continue for additional iterations after norm reaches
332: * stopping criterion.
333: *
334: IF( NRM.LT.DTPCRT )
335: $ GO TO 70
336: NRMCHK = NRMCHK + 1
337: IF( NRMCHK.LT.EXTRA+1 )
338: $ GO TO 70
339: *
340: GO TO 130
341: *
342: * If stopping criterion was not satisfied, update info and
343: * store eigenvector number in array ifail.
344: *
345: 120 CONTINUE
346: INFO = INFO + 1
347: IFAIL( INFO ) = J
348: *
349: * Accept iterate as jth eigenvector.
350: *
351: 130 CONTINUE
352: SCL = ONE / DNRM2( BLKSIZ, WORK( INDRV1+1 ), 1 )
353: JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
354: IF( WORK( INDRV1+JMAX ).LT.ZERO )
355: $ SCL = -SCL
356: CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
357: 140 CONTINUE
358: DO 150 I = 1, N
359: Z( I, J ) = CZERO
360: 150 CONTINUE
361: DO 160 I = 1, BLKSIZ
362: Z( B1+I-1, J ) = DCMPLX( WORK( INDRV1+I ), ZERO )
363: 160 CONTINUE
364: *
365: * Save the shift to check eigenvalue spacing at next
366: * iteration.
367: *
368: XJM = XJ
369: *
370: 170 CONTINUE
371: 180 CONTINUE
372: *
373: RETURN
374: *
375: * End of ZSTEIN
376: *
377: END
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