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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b SSTEBZ 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download SSTEDC + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sstedc.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sstedc.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sstedc.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE SSTEDC( 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 * REAL D( * ), E( * ), WORK( * ), Z( LDZ, * ) 00031 * .. 00032 * 00033 * 00034 *> \par Purpose: 00035 * ============= 00036 *> 00037 *> \verbatim 00038 *> 00039 *> SSTEDC 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 SSYTRD or SSPTRD or SSBTRD 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 SLAED3 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (MAX(1,LWORK)) 00109 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00110 *> \endverbatim 00111 *> 00112 *> \param[in] LWORK 00113 *> \verbatim 00114 *> LWORK is INTEGER 00115 *> The dimension of the array WORK. 00116 *> If COMPZ = 'N' or N <= 1 then LWORK must be at least 1. 00117 *> If COMPZ = 'V' and N > 1 then LWORK must be at least 00118 *> ( 1 + 3*N + 2*N*lg N + 4*N**2 ), 00119 *> where lg( N ) = smallest integer k such 00120 *> that 2**k >= N. 00121 *> If COMPZ = 'I' and N > 1 then LWORK must be at least 00122 *> ( 1 + 4*N + N**2 ). 00123 *> Note that for COMPZ = 'I' or 'V', then if N is less than or 00124 *> equal to the minimum divide size, usually 25, then LWORK need 00125 *> only be max(1,2*(N-1)). 00126 *> 00127 *> If LWORK = -1, then a workspace query is assumed; the routine 00128 *> only calculates the optimal size of the WORK array, returns 00129 *> this value as the first entry of the WORK array, and no error 00130 *> message related to LWORK is issued by XERBLA. 00131 *> \endverbatim 00132 *> 00133 *> \param[out] IWORK 00134 *> \verbatim 00135 *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) 00136 *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 00137 *> \endverbatim 00138 *> 00139 *> \param[in] LIWORK 00140 *> \verbatim 00141 *> LIWORK is INTEGER 00142 *> The dimension of the array IWORK. 00143 *> If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1. 00144 *> If COMPZ = 'V' and N > 1 then LIWORK must be at least 00145 *> ( 6 + 6*N + 5*N*lg N ). 00146 *> If COMPZ = 'I' and N > 1 then LIWORK must be at least 00147 *> ( 3 + 5*N ). 00148 *> Note that for COMPZ = 'I' or 'V', then if N is less than or 00149 *> equal to the minimum divide size, usually 25, then LIWORK 00150 *> need only be 1. 00151 *> 00152 *> If LIWORK = -1, then a workspace query is assumed; the 00153 *> routine only calculates the optimal size of the IWORK array, 00154 *> returns this value as the first entry of the IWORK array, and 00155 *> no error message related to LIWORK is issued by XERBLA. 00156 *> \endverbatim 00157 *> 00158 *> \param[out] INFO 00159 *> \verbatim 00160 *> INFO is INTEGER 00161 *> = 0: successful exit. 00162 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00163 *> > 0: The algorithm failed to compute an eigenvalue while 00164 *> working on the submatrix lying in rows and columns 00165 *> INFO/(N+1) through mod(INFO,N+1). 00166 *> \endverbatim 00167 * 00168 * Authors: 00169 * ======== 00170 * 00171 *> \author Univ. of Tennessee 00172 *> \author Univ. of California Berkeley 00173 *> \author Univ. of Colorado Denver 00174 *> \author NAG Ltd. 00175 * 00176 *> \date November 2011 00177 * 00178 *> \ingroup auxOTHERcomputational 00179 * 00180 *> \par Contributors: 00181 * ================== 00182 *> 00183 *> Jeff Rutter, Computer Science Division, University of California 00184 *> at Berkeley, USA \n 00185 *> Modified by Francoise Tisseur, University of Tennessee 00186 *> 00187 * ===================================================================== 00188 SUBROUTINE SSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, 00189 $ LIWORK, INFO ) 00190 * 00191 * -- LAPACK computational routine (version 3.4.0) -- 00192 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00193 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00194 * November 2011 00195 * 00196 * .. Scalar Arguments .. 00197 CHARACTER COMPZ 00198 INTEGER INFO, LDZ, LIWORK, LWORK, N 00199 * .. 00200 * .. Array Arguments .. 00201 INTEGER IWORK( * ) 00202 REAL D( * ), E( * ), WORK( * ), Z( LDZ, * ) 00203 * .. 00204 * 00205 * ===================================================================== 00206 * 00207 * .. Parameters .. 00208 REAL ZERO, ONE, TWO 00209 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0 ) 00210 * .. 00211 * .. Local Scalars .. 00212 LOGICAL LQUERY 00213 INTEGER FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN, 00214 $ LWMIN, M, SMLSIZ, START, STOREZ, STRTRW 00215 REAL EPS, ORGNRM, P, TINY 00216 * .. 00217 * .. External Functions .. 00218 LOGICAL LSAME 00219 INTEGER ILAENV 00220 REAL SLAMCH, SLANST 00221 EXTERNAL ILAENV, LSAME, SLAMCH, SLANST 00222 * .. 00223 * .. External Subroutines .. 00224 EXTERNAL SGEMM, SLACPY, SLAED0, SLASCL, SLASET, SLASRT, 00225 $ SSTEQR, SSTERF, SSWAP, XERBLA 00226 * .. 00227 * .. Intrinsic Functions .. 00228 INTRINSIC ABS, INT, LOG, MAX, MOD, REAL, SQRT 00229 * .. 00230 * .. Executable Statements .. 00231 * 00232 * Test the input parameters. 00233 * 00234 INFO = 0 00235 LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) 00236 * 00237 IF( LSAME( COMPZ, 'N' ) ) THEN 00238 ICOMPZ = 0 00239 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN 00240 ICOMPZ = 1 00241 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN 00242 ICOMPZ = 2 00243 ELSE 00244 ICOMPZ = -1 00245 END IF 00246 IF( ICOMPZ.LT.0 ) THEN 00247 INFO = -1 00248 ELSE IF( N.LT.0 ) THEN 00249 INFO = -2 00250 ELSE IF( ( LDZ.LT.1 ) .OR. 00251 $ ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, N ) ) ) THEN 00252 INFO = -6 00253 END IF 00254 * 00255 IF( INFO.EQ.0 ) THEN 00256 * 00257 * Compute the workspace requirements 00258 * 00259 SMLSIZ = ILAENV( 9, 'SSTEDC', ' ', 0, 0, 0, 0 ) 00260 IF( N.LE.1 .OR. ICOMPZ.EQ.0 ) THEN 00261 LIWMIN = 1 00262 LWMIN = 1 00263 ELSE IF( N.LE.SMLSIZ ) THEN 00264 LIWMIN = 1 00265 LWMIN = 2*( N - 1 ) 00266 ELSE 00267 LGN = INT( LOG( REAL( N ) )/LOG( TWO ) ) 00268 IF( 2**LGN.LT.N ) 00269 $ LGN = LGN + 1 00270 IF( 2**LGN.LT.N ) 00271 $ LGN = LGN + 1 00272 IF( ICOMPZ.EQ.1 ) THEN 00273 LWMIN = 1 + 3*N + 2*N*LGN + 4*N**2 00274 LIWMIN = 6 + 6*N + 5*N*LGN 00275 ELSE IF( ICOMPZ.EQ.2 ) THEN 00276 LWMIN = 1 + 4*N + N**2 00277 LIWMIN = 3 + 5*N 00278 END IF 00279 END IF 00280 WORK( 1 ) = LWMIN 00281 IWORK( 1 ) = LIWMIN 00282 * 00283 IF( LWORK.LT.LWMIN .AND. .NOT. LQUERY ) THEN 00284 INFO = -8 00285 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT. LQUERY ) THEN 00286 INFO = -10 00287 END IF 00288 END IF 00289 * 00290 IF( INFO.NE.0 ) THEN 00291 CALL XERBLA( 'SSTEDC', -INFO ) 00292 RETURN 00293 ELSE IF (LQUERY) THEN 00294 RETURN 00295 END IF 00296 * 00297 * Quick return if possible 00298 * 00299 IF( N.EQ.0 ) 00300 $ RETURN 00301 IF( N.EQ.1 ) THEN 00302 IF( ICOMPZ.NE.0 ) 00303 $ Z( 1, 1 ) = ONE 00304 RETURN 00305 END IF 00306 * 00307 * If the following conditional clause is removed, then the routine 00308 * will use the Divide and Conquer routine to compute only the 00309 * eigenvalues, which requires (3N + 3N**2) real workspace and 00310 * (2 + 5N + 2N lg(N)) integer workspace. 00311 * Since on many architectures SSTERF is much faster than any other 00312 * algorithm for finding eigenvalues only, it is used here 00313 * as the default. If the conditional clause is removed, then 00314 * information on the size of workspace needs to be changed. 00315 * 00316 * If COMPZ = 'N', use SSTERF to compute the eigenvalues. 00317 * 00318 IF( ICOMPZ.EQ.0 ) THEN 00319 CALL SSTERF( N, D, E, INFO ) 00320 GO TO 50 00321 END IF 00322 * 00323 * If N is smaller than the minimum divide size (SMLSIZ+1), then 00324 * solve the problem with another solver. 00325 * 00326 IF( N.LE.SMLSIZ ) THEN 00327 * 00328 CALL SSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) 00329 * 00330 ELSE 00331 * 00332 * If COMPZ = 'V', the Z matrix must be stored elsewhere for later 00333 * use. 00334 * 00335 IF( ICOMPZ.EQ.1 ) THEN 00336 STOREZ = 1 + N*N 00337 ELSE 00338 STOREZ = 1 00339 END IF 00340 * 00341 IF( ICOMPZ.EQ.2 ) THEN 00342 CALL SLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) 00343 END IF 00344 * 00345 * Scale. 00346 * 00347 ORGNRM = SLANST( 'M', N, D, E ) 00348 IF( ORGNRM.EQ.ZERO ) 00349 $ GO TO 50 00350 * 00351 EPS = SLAMCH( 'Epsilon' ) 00352 * 00353 START = 1 00354 * 00355 * while ( START <= N ) 00356 * 00357 10 CONTINUE 00358 IF( START.LE.N ) THEN 00359 * 00360 * Let FINISH be the position of the next subdiagonal entry 00361 * such that E( FINISH ) <= TINY or FINISH = N if no such 00362 * subdiagonal exists. The matrix identified by the elements 00363 * between START and FINISH constitutes an independent 00364 * sub-problem. 00365 * 00366 FINISH = START 00367 20 CONTINUE 00368 IF( FINISH.LT.N ) THEN 00369 TINY = EPS*SQRT( ABS( D( FINISH ) ) )* 00370 $ SQRT( ABS( D( FINISH+1 ) ) ) 00371 IF( ABS( E( FINISH ) ).GT.TINY ) THEN 00372 FINISH = FINISH + 1 00373 GO TO 20 00374 END IF 00375 END IF 00376 * 00377 * (Sub) Problem determined. Compute its size and solve it. 00378 * 00379 M = FINISH - START + 1 00380 IF( M.EQ.1 ) THEN 00381 START = FINISH + 1 00382 GO TO 10 00383 END IF 00384 IF( M.GT.SMLSIZ ) THEN 00385 * 00386 * Scale. 00387 * 00388 ORGNRM = SLANST( 'M', M, D( START ), E( START ) ) 00389 CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ), M, 00390 $ INFO ) 00391 CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1, E( START ), 00392 $ M-1, INFO ) 00393 * 00394 IF( ICOMPZ.EQ.1 ) THEN 00395 STRTRW = 1 00396 ELSE 00397 STRTRW = START 00398 END IF 00399 CALL SLAED0( ICOMPZ, N, M, D( START ), E( START ), 00400 $ Z( STRTRW, START ), LDZ, WORK( 1 ), N, 00401 $ WORK( STOREZ ), IWORK, INFO ) 00402 IF( INFO.NE.0 ) THEN 00403 INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) + 00404 $ MOD( INFO, ( M+1 ) ) + START - 1 00405 GO TO 50 00406 END IF 00407 * 00408 * Scale back. 00409 * 00410 CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ), M, 00411 $ INFO ) 00412 * 00413 ELSE 00414 IF( ICOMPZ.EQ.1 ) THEN 00415 * 00416 * Since QR won't update a Z matrix which is larger than 00417 * the length of D, we must solve the sub-problem in a 00418 * workspace and then multiply back into Z. 00419 * 00420 CALL SSTEQR( 'I', M, D( START ), E( START ), WORK, M, 00421 $ WORK( M*M+1 ), INFO ) 00422 CALL SLACPY( 'A', N, M, Z( 1, START ), LDZ, 00423 $ WORK( STOREZ ), N ) 00424 CALL SGEMM( 'N', 'N', N, M, M, ONE, 00425 $ WORK( STOREZ ), N, WORK, M, ZERO, 00426 $ Z( 1, START ), LDZ ) 00427 ELSE IF( ICOMPZ.EQ.2 ) THEN 00428 CALL SSTEQR( 'I', M, D( START ), E( START ), 00429 $ Z( START, START ), LDZ, WORK, INFO ) 00430 ELSE 00431 CALL SSTERF( M, D( START ), E( START ), INFO ) 00432 END IF 00433 IF( INFO.NE.0 ) THEN 00434 INFO = START*( N+1 ) + FINISH 00435 GO TO 50 00436 END IF 00437 END IF 00438 * 00439 START = FINISH + 1 00440 GO TO 10 00441 END IF 00442 * 00443 * endwhile 00444 * 00445 * If the problem split any number of times, then the eigenvalues 00446 * will not be properly ordered. Here we permute the eigenvalues 00447 * (and the associated eigenvectors) into ascending order. 00448 * 00449 IF( M.NE.N ) THEN 00450 IF( ICOMPZ.EQ.0 ) THEN 00451 * 00452 * Use Quick Sort 00453 * 00454 CALL SLASRT( 'I', N, D, INFO ) 00455 * 00456 ELSE 00457 * 00458 * Use Selection Sort to minimize swaps of eigenvectors 00459 * 00460 DO 40 II = 2, N 00461 I = II - 1 00462 K = I 00463 P = D( I ) 00464 DO 30 J = II, N 00465 IF( D( J ).LT.P ) THEN 00466 K = J 00467 P = D( J ) 00468 END IF 00469 30 CONTINUE 00470 IF( K.NE.I ) THEN 00471 D( K ) = D( I ) 00472 D( I ) = P 00473 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 ) 00474 END IF 00475 40 CONTINUE 00476 END IF 00477 END IF 00478 END IF 00479 * 00480 50 CONTINUE 00481 WORK( 1 ) = LWMIN 00482 IWORK( 1 ) = LIWMIN 00483 * 00484 RETURN 00485 * 00486 * End of SSTEDC 00487 * 00488 END