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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DSTEBZ 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DSTEDC + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstedc.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstedc.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstedc.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, 00022 * LIWORK, INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * CHARACTER COMPZ 00026 * INTEGER INFO, LDZ, LIWORK, LWORK, N 00027 * .. 00028 * .. Array Arguments .. 00029 * INTEGER IWORK( * ) 00030 * DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * ) 00031 * .. 00032 * 00033 * 00034 *> \par Purpose: 00035 * ============= 00036 *> 00037 *> \verbatim 00038 *> 00039 *> DSTEDC computes all eigenvalues and, optionally, eigenvectors of a 00040 *> symmetric tridiagonal matrix using the divide and conquer method. 00041 *> The eigenvectors of a full or band real symmetric matrix can also be 00042 *> found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this 00043 *> matrix to tridiagonal form. 00044 *> 00045 *> This code makes very mild assumptions about floating point 00046 *> arithmetic. It will work on machines with a guard digit in 00047 *> add/subtract, or on those binary machines without guard digits 00048 *> which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. 00049 *> It could conceivably fail on hexadecimal or decimal machines 00050 *> without guard digits, but we know of none. See DLAED3 for details. 00051 *> \endverbatim 00052 * 00053 * Arguments: 00054 * ========== 00055 * 00056 *> \param[in] COMPZ 00057 *> \verbatim 00058 *> COMPZ is CHARACTER*1 00059 *> = 'N': Compute eigenvalues only. 00060 *> = 'I': Compute eigenvectors of tridiagonal matrix also. 00061 *> = 'V': Compute eigenvectors of original dense symmetric 00062 *> matrix also. On entry, Z contains the orthogonal 00063 *> matrix used to reduce the original matrix to 00064 *> tridiagonal form. 00065 *> \endverbatim 00066 *> 00067 *> \param[in] N 00068 *> \verbatim 00069 *> N is INTEGER 00070 *> The dimension of the symmetric tridiagonal matrix. N >= 0. 00071 *> \endverbatim 00072 *> 00073 *> \param[in,out] D 00074 *> \verbatim 00075 *> D is DOUBLE PRECISION array, dimension (N) 00076 *> On entry, the diagonal elements of the tridiagonal matrix. 00077 *> On exit, if INFO = 0, the eigenvalues in ascending order. 00078 *> \endverbatim 00079 *> 00080 *> \param[in,out] E 00081 *> \verbatim 00082 *> E is DOUBLE PRECISION array, dimension (N-1) 00083 *> On entry, the subdiagonal elements of the tridiagonal matrix. 00084 *> On exit, E has been destroyed. 00085 *> \endverbatim 00086 *> 00087 *> \param[in,out] Z 00088 *> \verbatim 00089 *> Z is DOUBLE PRECISION array, dimension (LDZ,N) 00090 *> On entry, if COMPZ = 'V', then Z contains the orthogonal 00091 *> matrix used in the reduction to tridiagonal form. 00092 *> On exit, if INFO = 0, then if COMPZ = 'V', Z contains the 00093 *> orthonormal eigenvectors of the original symmetric matrix, 00094 *> and if COMPZ = 'I', Z contains the orthonormal eigenvectors 00095 *> of the symmetric tridiagonal matrix. 00096 *> If COMPZ = 'N', then Z is not referenced. 00097 *> \endverbatim 00098 *> 00099 *> \param[in] LDZ 00100 *> \verbatim 00101 *> LDZ is INTEGER 00102 *> The leading dimension of the array Z. LDZ >= 1. 00103 *> If eigenvectors are desired, then LDZ >= max(1,N). 00104 *> \endverbatim 00105 *> 00106 *> \param[out] WORK 00107 *> \verbatim 00108 *> WORK is DOUBLE PRECISION array, 00109 *> dimension (LWORK) 00110 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00111 *> \endverbatim 00112 *> 00113 *> \param[in] LWORK 00114 *> \verbatim 00115 *> LWORK is INTEGER 00116 *> The dimension of the array WORK. 00117 *> If COMPZ = 'N' or N <= 1 then LWORK must be at least 1. 00118 *> If COMPZ = 'V' and N > 1 then LWORK must be at least 00119 *> ( 1 + 3*N + 2*N*lg N + 4*N**2 ), 00120 *> where lg( N ) = smallest integer k such 00121 *> that 2**k >= N. 00122 *> If COMPZ = 'I' and N > 1 then LWORK must be at least 00123 *> ( 1 + 4*N + N**2 ). 00124 *> Note that for COMPZ = 'I' or 'V', then if N is less than or 00125 *> equal to the minimum divide size, usually 25, then LWORK need 00126 *> only be max(1,2*(N-1)). 00127 *> 00128 *> If LWORK = -1, then a workspace query is assumed; the routine 00129 *> only calculates the optimal size of the WORK array, returns 00130 *> this value as the first entry of the WORK array, and no error 00131 *> message related to LWORK is issued by XERBLA. 00132 *> \endverbatim 00133 *> 00134 *> \param[out] IWORK 00135 *> \verbatim 00136 *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) 00137 *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 00138 *> \endverbatim 00139 *> 00140 *> \param[in] LIWORK 00141 *> \verbatim 00142 *> LIWORK is INTEGER 00143 *> The dimension of the array IWORK. 00144 *> If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1. 00145 *> If COMPZ = 'V' and N > 1 then LIWORK must be at least 00146 *> ( 6 + 6*N + 5*N*lg N ). 00147 *> If COMPZ = 'I' and N > 1 then LIWORK must be at least 00148 *> ( 3 + 5*N ). 00149 *> Note that for COMPZ = 'I' or 'V', then if N is less than or 00150 *> equal to the minimum divide size, usually 25, then LIWORK 00151 *> need only be 1. 00152 *> 00153 *> If LIWORK = -1, then a workspace query is assumed; the 00154 *> routine only calculates the optimal size of the IWORK array, 00155 *> returns this value as the first entry of the IWORK array, and 00156 *> no error message related to LIWORK is issued by XERBLA. 00157 *> \endverbatim 00158 *> 00159 *> \param[out] INFO 00160 *> \verbatim 00161 *> INFO is INTEGER 00162 *> = 0: successful exit. 00163 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00164 *> > 0: The algorithm failed to compute an eigenvalue while 00165 *> working on the submatrix lying in rows and columns 00166 *> INFO/(N+1) through mod(INFO,N+1). 00167 *> \endverbatim 00168 * 00169 * Authors: 00170 * ======== 00171 * 00172 *> \author Univ. of Tennessee 00173 *> \author Univ. of California Berkeley 00174 *> \author Univ. of Colorado Denver 00175 *> \author NAG Ltd. 00176 * 00177 *> \date November 2011 00178 * 00179 *> \ingroup auxOTHERcomputational 00180 * 00181 *> \par Contributors: 00182 * ================== 00183 *> 00184 *> Jeff Rutter, Computer Science Division, University of California 00185 *> at Berkeley, USA \n 00186 *> Modified by Francoise Tisseur, University of Tennessee 00187 *> 00188 * ===================================================================== 00189 SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, 00190 $ LIWORK, INFO ) 00191 * 00192 * -- LAPACK computational routine (version 3.4.0) -- 00193 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00195 * November 2011 00196 * 00197 * .. Scalar Arguments .. 00198 CHARACTER COMPZ 00199 INTEGER INFO, LDZ, LIWORK, LWORK, N 00200 * .. 00201 * .. Array Arguments .. 00202 INTEGER IWORK( * ) 00203 DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * ) 00204 * .. 00205 * 00206 * ===================================================================== 00207 * 00208 * .. Parameters .. 00209 DOUBLE PRECISION ZERO, ONE, TWO 00210 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 ) 00211 * .. 00212 * .. Local Scalars .. 00213 LOGICAL LQUERY 00214 INTEGER FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN, 00215 $ LWMIN, M, SMLSIZ, START, STOREZ, STRTRW 00216 DOUBLE PRECISION EPS, ORGNRM, P, TINY 00217 * .. 00218 * .. External Functions .. 00219 LOGICAL LSAME 00220 INTEGER ILAENV 00221 DOUBLE PRECISION DLAMCH, DLANST 00222 EXTERNAL LSAME, ILAENV, DLAMCH, DLANST 00223 * .. 00224 * .. External Subroutines .. 00225 EXTERNAL DGEMM, DLACPY, DLAED0, DLASCL, DLASET, DLASRT, 00226 $ DSTEQR, DSTERF, DSWAP, XERBLA 00227 * .. 00228 * .. Intrinsic Functions .. 00229 INTRINSIC ABS, DBLE, INT, LOG, MAX, MOD, SQRT 00230 * .. 00231 * .. Executable Statements .. 00232 * 00233 * Test the input parameters. 00234 * 00235 INFO = 0 00236 LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) 00237 * 00238 IF( LSAME( COMPZ, 'N' ) ) THEN 00239 ICOMPZ = 0 00240 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN 00241 ICOMPZ = 1 00242 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN 00243 ICOMPZ = 2 00244 ELSE 00245 ICOMPZ = -1 00246 END IF 00247 IF( ICOMPZ.LT.0 ) THEN 00248 INFO = -1 00249 ELSE IF( N.LT.0 ) THEN 00250 INFO = -2 00251 ELSE IF( ( LDZ.LT.1 ) .OR. 00252 $ ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, N ) ) ) THEN 00253 INFO = -6 00254 END IF 00255 * 00256 IF( INFO.EQ.0 ) THEN 00257 * 00258 * Compute the workspace requirements 00259 * 00260 SMLSIZ = ILAENV( 9, 'DSTEDC', ' ', 0, 0, 0, 0 ) 00261 IF( N.LE.1 .OR. ICOMPZ.EQ.0 ) THEN 00262 LIWMIN = 1 00263 LWMIN = 1 00264 ELSE IF( N.LE.SMLSIZ ) THEN 00265 LIWMIN = 1 00266 LWMIN = 2*( N - 1 ) 00267 ELSE 00268 LGN = INT( LOG( DBLE( N ) )/LOG( TWO ) ) 00269 IF( 2**LGN.LT.N ) 00270 $ LGN = LGN + 1 00271 IF( 2**LGN.LT.N ) 00272 $ LGN = LGN + 1 00273 IF( ICOMPZ.EQ.1 ) THEN 00274 LWMIN = 1 + 3*N + 2*N*LGN + 4*N**2 00275 LIWMIN = 6 + 6*N + 5*N*LGN 00276 ELSE IF( ICOMPZ.EQ.2 ) THEN 00277 LWMIN = 1 + 4*N + N**2 00278 LIWMIN = 3 + 5*N 00279 END IF 00280 END IF 00281 WORK( 1 ) = LWMIN 00282 IWORK( 1 ) = LIWMIN 00283 * 00284 IF( LWORK.LT.LWMIN .AND. .NOT. LQUERY ) THEN 00285 INFO = -8 00286 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT. LQUERY ) THEN 00287 INFO = -10 00288 END IF 00289 END IF 00290 * 00291 IF( INFO.NE.0 ) THEN 00292 CALL XERBLA( 'DSTEDC', -INFO ) 00293 RETURN 00294 ELSE IF (LQUERY) THEN 00295 RETURN 00296 END IF 00297 * 00298 * Quick return if possible 00299 * 00300 IF( N.EQ.0 ) 00301 $ RETURN 00302 IF( N.EQ.1 ) THEN 00303 IF( ICOMPZ.NE.0 ) 00304 $ Z( 1, 1 ) = ONE 00305 RETURN 00306 END IF 00307 * 00308 * If the following conditional clause is removed, then the routine 00309 * will use the Divide and Conquer routine to compute only the 00310 * eigenvalues, which requires (3N + 3N**2) real workspace and 00311 * (2 + 5N + 2N lg(N)) integer workspace. 00312 * Since on many architectures DSTERF is much faster than any other 00313 * algorithm for finding eigenvalues only, it is used here 00314 * as the default. If the conditional clause is removed, then 00315 * information on the size of workspace needs to be changed. 00316 * 00317 * If COMPZ = 'N', use DSTERF to compute the eigenvalues. 00318 * 00319 IF( ICOMPZ.EQ.0 ) THEN 00320 CALL DSTERF( N, D, E, INFO ) 00321 GO TO 50 00322 END IF 00323 * 00324 * If N is smaller than the minimum divide size (SMLSIZ+1), then 00325 * solve the problem with another solver. 00326 * 00327 IF( N.LE.SMLSIZ ) THEN 00328 * 00329 CALL DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) 00330 * 00331 ELSE 00332 * 00333 * If COMPZ = 'V', the Z matrix must be stored elsewhere for later 00334 * use. 00335 * 00336 IF( ICOMPZ.EQ.1 ) THEN 00337 STOREZ = 1 + N*N 00338 ELSE 00339 STOREZ = 1 00340 END IF 00341 * 00342 IF( ICOMPZ.EQ.2 ) THEN 00343 CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) 00344 END IF 00345 * 00346 * Scale. 00347 * 00348 ORGNRM = DLANST( 'M', N, D, E ) 00349 IF( ORGNRM.EQ.ZERO ) 00350 $ GO TO 50 00351 * 00352 EPS = DLAMCH( 'Epsilon' ) 00353 * 00354 START = 1 00355 * 00356 * while ( START <= N ) 00357 * 00358 10 CONTINUE 00359 IF( START.LE.N ) THEN 00360 * 00361 * Let FINISH be the position of the next subdiagonal entry 00362 * such that E( FINISH ) <= TINY or FINISH = N if no such 00363 * subdiagonal exists. The matrix identified by the elements 00364 * between START and FINISH constitutes an independent 00365 * sub-problem. 00366 * 00367 FINISH = START 00368 20 CONTINUE 00369 IF( FINISH.LT.N ) THEN 00370 TINY = EPS*SQRT( ABS( D( FINISH ) ) )* 00371 $ SQRT( ABS( D( FINISH+1 ) ) ) 00372 IF( ABS( E( FINISH ) ).GT.TINY ) THEN 00373 FINISH = FINISH + 1 00374 GO TO 20 00375 END IF 00376 END IF 00377 * 00378 * (Sub) Problem determined. Compute its size and solve it. 00379 * 00380 M = FINISH - START + 1 00381 IF( M.EQ.1 ) THEN 00382 START = FINISH + 1 00383 GO TO 10 00384 END IF 00385 IF( M.GT.SMLSIZ ) THEN 00386 * 00387 * Scale. 00388 * 00389 ORGNRM = DLANST( 'M', M, D( START ), E( START ) ) 00390 CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ), M, 00391 $ INFO ) 00392 CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1, E( START ), 00393 $ M-1, INFO ) 00394 * 00395 IF( ICOMPZ.EQ.1 ) THEN 00396 STRTRW = 1 00397 ELSE 00398 STRTRW = START 00399 END IF 00400 CALL DLAED0( ICOMPZ, N, M, D( START ), E( START ), 00401 $ Z( STRTRW, START ), LDZ, WORK( 1 ), N, 00402 $ WORK( STOREZ ), IWORK, INFO ) 00403 IF( INFO.NE.0 ) THEN 00404 INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) + 00405 $ MOD( INFO, ( M+1 ) ) + START - 1 00406 GO TO 50 00407 END IF 00408 * 00409 * Scale back. 00410 * 00411 CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ), M, 00412 $ INFO ) 00413 * 00414 ELSE 00415 IF( ICOMPZ.EQ.1 ) THEN 00416 * 00417 * Since QR won't update a Z matrix which is larger than 00418 * the length of D, we must solve the sub-problem in a 00419 * workspace and then multiply back into Z. 00420 * 00421 CALL DSTEQR( 'I', M, D( START ), E( START ), WORK, M, 00422 $ WORK( M*M+1 ), INFO ) 00423 CALL DLACPY( 'A', N, M, Z( 1, START ), LDZ, 00424 $ WORK( STOREZ ), N ) 00425 CALL DGEMM( 'N', 'N', N, M, M, ONE, 00426 $ WORK( STOREZ ), N, WORK, M, ZERO, 00427 $ Z( 1, START ), LDZ ) 00428 ELSE IF( ICOMPZ.EQ.2 ) THEN 00429 CALL DSTEQR( 'I', M, D( START ), E( START ), 00430 $ Z( START, START ), LDZ, WORK, INFO ) 00431 ELSE 00432 CALL DSTERF( M, D( START ), E( START ), INFO ) 00433 END IF 00434 IF( INFO.NE.0 ) THEN 00435 INFO = START*( N+1 ) + FINISH 00436 GO TO 50 00437 END IF 00438 END IF 00439 * 00440 START = FINISH + 1 00441 GO TO 10 00442 END IF 00443 * 00444 * endwhile 00445 * 00446 * If the problem split any number of times, then the eigenvalues 00447 * will not be properly ordered. Here we permute the eigenvalues 00448 * (and the associated eigenvectors) into ascending order. 00449 * 00450 IF( M.NE.N ) THEN 00451 IF( ICOMPZ.EQ.0 ) THEN 00452 * 00453 * Use Quick Sort 00454 * 00455 CALL DLASRT( 'I', N, D, INFO ) 00456 * 00457 ELSE 00458 * 00459 * Use Selection Sort to minimize swaps of eigenvectors 00460 * 00461 DO 40 II = 2, N 00462 I = II - 1 00463 K = I 00464 P = D( I ) 00465 DO 30 J = II, N 00466 IF( D( J ).LT.P ) THEN 00467 K = J 00468 P = D( J ) 00469 END IF 00470 30 CONTINUE 00471 IF( K.NE.I ) THEN 00472 D( K ) = D( I ) 00473 D( I ) = P 00474 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 ) 00475 END IF 00476 40 CONTINUE 00477 END IF 00478 END IF 00479 END IF 00480 * 00481 50 CONTINUE 00482 WORK( 1 ) = LWMIN 00483 IWORK( 1 ) = LIWMIN 00484 * 00485 RETURN 00486 * 00487 * End of DSTEDC 00488 * 00489 END