1: SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
2: $ ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
3: $ INFO )
4: *
5: * -- LAPACK driver 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: CHARACTER JOBZ, RANGE, UPLO
12: INTEGER IL, INFO, IU, LDZ, M, N
13: DOUBLE PRECISION ABSTOL, VL, VU
14: * ..
15: * .. Array Arguments ..
16: INTEGER IFAIL( * ), IWORK( * )
17: DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
18: * ..
19: *
20: * Purpose
21: * =======
22: *
23: * DSPEVX computes selected eigenvalues and, optionally, eigenvectors
24: * of a real symmetric matrix A in packed storage. Eigenvalues/vectors
25: * can be selected by specifying either a range of values or a range of
26: * indices for the desired eigenvalues.
27: *
28: * Arguments
29: * =========
30: *
31: * JOBZ (input) CHARACTER*1
32: * = 'N': Compute eigenvalues only;
33: * = 'V': Compute eigenvalues and eigenvectors.
34: *
35: * RANGE (input) CHARACTER*1
36: * = 'A': all eigenvalues will be found;
37: * = 'V': all eigenvalues in the half-open interval (VL,VU]
38: * will be found;
39: * = 'I': the IL-th through IU-th eigenvalues will be found.
40: *
41: * UPLO (input) CHARACTER*1
42: * = 'U': Upper triangle of A is stored;
43: * = 'L': Lower triangle of A is stored.
44: *
45: * N (input) INTEGER
46: * The order of the matrix A. N >= 0.
47: *
48: * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
49: * On entry, the upper or lower triangle of the symmetric matrix
50: * A, packed columnwise in a linear array. The j-th column of A
51: * is stored in the array AP as follows:
52: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
53: * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
54: *
55: * On exit, AP is overwritten by values generated during the
56: * reduction to tridiagonal form. If UPLO = 'U', the diagonal
57: * and first superdiagonal of the tridiagonal matrix T overwrite
58: * the corresponding elements of A, and if UPLO = 'L', the
59: * diagonal and first subdiagonal of T overwrite the
60: * corresponding elements of A.
61: *
62: * VL (input) DOUBLE PRECISION
63: * VU (input) DOUBLE PRECISION
64: * If RANGE='V', the lower and upper bounds of the interval to
65: * be searched for eigenvalues. VL < VU.
66: * Not referenced if RANGE = 'A' or 'I'.
67: *
68: * IL (input) INTEGER
69: * IU (input) INTEGER
70: * If RANGE='I', the indices (in ascending order) of the
71: * smallest and largest eigenvalues to be returned.
72: * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
73: * Not referenced if RANGE = 'A' or 'V'.
74: *
75: * ABSTOL (input) DOUBLE PRECISION
76: * The absolute error tolerance for the eigenvalues.
77: * An approximate eigenvalue is accepted as converged
78: * when it is determined to lie in an interval [a,b]
79: * of width less than or equal to
80: *
81: * ABSTOL + EPS * max( |a|,|b| ) ,
82: *
83: * where EPS is the machine precision. If ABSTOL is less than
84: * or equal to zero, then EPS*|T| will be used in its place,
85: * where |T| is the 1-norm of the tridiagonal matrix obtained
86: * by reducing AP to tridiagonal form.
87: *
88: * Eigenvalues will be computed most accurately when ABSTOL is
89: * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
90: * If this routine returns with INFO>0, indicating that some
91: * eigenvectors did not converge, try setting ABSTOL to
92: * 2*DLAMCH('S').
93: *
94: * See "Computing Small Singular Values of Bidiagonal Matrices
95: * with Guaranteed High Relative Accuracy," by Demmel and
96: * Kahan, LAPACK Working Note #3.
97: *
98: * M (output) INTEGER
99: * The total number of eigenvalues found. 0 <= M <= N.
100: * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
101: *
102: * W (output) DOUBLE PRECISION array, dimension (N)
103: * If INFO = 0, the selected eigenvalues in ascending order.
104: *
105: * Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
106: * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
107: * contain the orthonormal eigenvectors of the matrix A
108: * corresponding to the selected eigenvalues, with the i-th
109: * column of Z holding the eigenvector associated with W(i).
110: * If an eigenvector fails to converge, then that column of Z
111: * contains the latest approximation to the eigenvector, and the
112: * index of the eigenvector is returned in IFAIL.
113: * If JOBZ = 'N', then Z is not referenced.
114: * Note: the user must ensure that at least max(1,M) columns are
115: * supplied in the array Z; if RANGE = 'V', the exact value of M
116: * is not known in advance and an upper bound must be used.
117: *
118: * LDZ (input) INTEGER
119: * The leading dimension of the array Z. LDZ >= 1, and if
120: * JOBZ = 'V', LDZ >= max(1,N).
121: *
122: * WORK (workspace) DOUBLE PRECISION array, dimension (8*N)
123: *
124: * IWORK (workspace) INTEGER array, dimension (5*N)
125: *
126: * IFAIL (output) INTEGER array, dimension (N)
127: * If JOBZ = 'V', then if INFO = 0, the first M elements of
128: * IFAIL are zero. If INFO > 0, then IFAIL contains the
129: * indices of the eigenvectors that failed to converge.
130: * If JOBZ = 'N', then IFAIL is not referenced.
131: *
132: * INFO (output) INTEGER
133: * = 0: successful exit
134: * < 0: if INFO = -i, the i-th argument had an illegal value
135: * > 0: if INFO = i, then i eigenvectors failed to converge.
136: * Their indices are stored in array IFAIL.
137: *
138: * =====================================================================
139: *
140: * .. Parameters ..
141: DOUBLE PRECISION ZERO, ONE
142: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
143: * ..
144: * .. Local Scalars ..
145: LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
146: CHARACTER ORDER
147: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
148: $ INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1,
149: $ J, JJ, NSPLIT
150: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
151: $ SIGMA, SMLNUM, TMP1, VLL, VUU
152: * ..
153: * .. External Functions ..
154: LOGICAL LSAME
155: DOUBLE PRECISION DLAMCH, DLANSP
156: EXTERNAL LSAME, DLAMCH, DLANSP
157: * ..
158: * .. External Subroutines ..
159: EXTERNAL DCOPY, DOPGTR, DOPMTR, DSCAL, DSPTRD, DSTEBZ,
160: $ DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
161: * ..
162: * .. Intrinsic Functions ..
163: INTRINSIC MAX, MIN, SQRT
164: * ..
165: * .. Executable Statements ..
166: *
167: * Test the input parameters.
168: *
169: WANTZ = LSAME( JOBZ, 'V' )
170: ALLEIG = LSAME( RANGE, 'A' )
171: VALEIG = LSAME( RANGE, 'V' )
172: INDEIG = LSAME( RANGE, 'I' )
173: *
174: INFO = 0
175: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
176: INFO = -1
177: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
178: INFO = -2
179: ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
180: $ THEN
181: INFO = -3
182: ELSE IF( N.LT.0 ) THEN
183: INFO = -4
184: ELSE
185: IF( VALEIG ) THEN
186: IF( N.GT.0 .AND. VU.LE.VL )
187: $ INFO = -7
188: ELSE IF( INDEIG ) THEN
189: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
190: INFO = -8
191: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
192: INFO = -9
193: END IF
194: END IF
195: END IF
196: IF( INFO.EQ.0 ) THEN
197: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
198: $ INFO = -14
199: END IF
200: *
201: IF( INFO.NE.0 ) THEN
202: CALL XERBLA( 'DSPEVX', -INFO )
203: RETURN
204: END IF
205: *
206: * Quick return if possible
207: *
208: M = 0
209: IF( N.EQ.0 )
210: $ RETURN
211: *
212: IF( N.EQ.1 ) THEN
213: IF( ALLEIG .OR. INDEIG ) THEN
214: M = 1
215: W( 1 ) = AP( 1 )
216: ELSE
217: IF( VL.LT.AP( 1 ) .AND. VU.GE.AP( 1 ) ) THEN
218: M = 1
219: W( 1 ) = AP( 1 )
220: END IF
221: END IF
222: IF( WANTZ )
223: $ Z( 1, 1 ) = ONE
224: RETURN
225: END IF
226: *
227: * Get machine constants.
228: *
229: SAFMIN = DLAMCH( 'Safe minimum' )
230: EPS = DLAMCH( 'Precision' )
231: SMLNUM = SAFMIN / EPS
232: BIGNUM = ONE / SMLNUM
233: RMIN = SQRT( SMLNUM )
234: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
235: *
236: * Scale matrix to allowable range, if necessary.
237: *
238: ISCALE = 0
239: ABSTLL = ABSTOL
240: IF( VALEIG ) THEN
241: VLL = VL
242: VUU = VU
243: ELSE
244: VLL = ZERO
245: VUU = ZERO
246: END IF
247: ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
248: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
249: ISCALE = 1
250: SIGMA = RMIN / ANRM
251: ELSE IF( ANRM.GT.RMAX ) THEN
252: ISCALE = 1
253: SIGMA = RMAX / ANRM
254: END IF
255: IF( ISCALE.EQ.1 ) THEN
256: CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
257: IF( ABSTOL.GT.0 )
258: $ ABSTLL = ABSTOL*SIGMA
259: IF( VALEIG ) THEN
260: VLL = VL*SIGMA
261: VUU = VU*SIGMA
262: END IF
263: END IF
264: *
265: * Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
266: *
267: INDTAU = 1
268: INDE = INDTAU + N
269: INDD = INDE + N
270: INDWRK = INDD + N
271: CALL DSPTRD( UPLO, N, AP, WORK( INDD ), WORK( INDE ),
272: $ WORK( INDTAU ), IINFO )
273: *
274: * If all eigenvalues are desired and ABSTOL is less than or equal
275: * to zero, then call DSTERF or DOPGTR and SSTEQR. If this fails
276: * for some eigenvalue, then try DSTEBZ.
277: *
278: TEST = .FALSE.
279: IF (INDEIG) THEN
280: IF (IL.EQ.1 .AND. IU.EQ.N) THEN
281: TEST = .TRUE.
282: END IF
283: END IF
284: IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
285: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
286: INDEE = INDWRK + 2*N
287: IF( .NOT.WANTZ ) THEN
288: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
289: CALL DSTERF( N, W, WORK( INDEE ), INFO )
290: ELSE
291: CALL DOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
292: $ WORK( INDWRK ), IINFO )
293: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
294: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
295: $ WORK( INDWRK ), INFO )
296: IF( INFO.EQ.0 ) THEN
297: DO 10 I = 1, N
298: IFAIL( I ) = 0
299: 10 CONTINUE
300: END IF
301: END IF
302: IF( INFO.EQ.0 ) THEN
303: M = N
304: GO TO 20
305: END IF
306: INFO = 0
307: END IF
308: *
309: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
310: *
311: IF( WANTZ ) THEN
312: ORDER = 'B'
313: ELSE
314: ORDER = 'E'
315: END IF
316: INDIBL = 1
317: INDISP = INDIBL + N
318: INDIWO = INDISP + N
319: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
320: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
321: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
322: $ IWORK( INDIWO ), INFO )
323: *
324: IF( WANTZ ) THEN
325: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
326: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
327: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
328: *
329: * Apply orthogonal matrix used in reduction to tridiagonal
330: * form to eigenvectors returned by DSTEIN.
331: *
332: CALL DOPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ,
333: $ WORK( INDWRK ), IINFO )
334: END IF
335: *
336: * If matrix was scaled, then rescale eigenvalues appropriately.
337: *
338: 20 CONTINUE
339: IF( ISCALE.EQ.1 ) THEN
340: IF( INFO.EQ.0 ) THEN
341: IMAX = M
342: ELSE
343: IMAX = INFO - 1
344: END IF
345: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
346: END IF
347: *
348: * If eigenvalues are not in order, then sort them, along with
349: * eigenvectors.
350: *
351: IF( WANTZ ) THEN
352: DO 40 J = 1, M - 1
353: I = 0
354: TMP1 = W( J )
355: DO 30 JJ = J + 1, M
356: IF( W( JJ ).LT.TMP1 ) THEN
357: I = JJ
358: TMP1 = W( JJ )
359: END IF
360: 30 CONTINUE
361: *
362: IF( I.NE.0 ) THEN
363: ITMP1 = IWORK( INDIBL+I-1 )
364: W( I ) = W( J )
365: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
366: W( J ) = TMP1
367: IWORK( INDIBL+J-1 ) = ITMP1
368: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
369: IF( INFO.NE.0 ) THEN
370: ITMP1 = IFAIL( I )
371: IFAIL( I ) = IFAIL( J )
372: IFAIL( J ) = ITMP1
373: END IF
374: END IF
375: 40 CONTINUE
376: END IF
377: *
378: RETURN
379: *
380: * End of DSPEVX
381: *
382: END
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