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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE SSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, 00002 $ LIWORK, INFO ) 00003 * 00004 * -- LAPACK driver routine (version 3.2) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * November 2006 00008 * 00009 * .. Scalar Arguments .. 00010 CHARACTER COMPZ 00011 INTEGER INFO, LDZ, LIWORK, LWORK, N 00012 * .. 00013 * .. Array Arguments .. 00014 INTEGER IWORK( * ) 00015 REAL D( * ), E( * ), WORK( * ), Z( LDZ, * ) 00016 * .. 00017 * 00018 * Purpose 00019 * ======= 00020 * 00021 * SSTEDC computes all eigenvalues and, optionally, eigenvectors of a 00022 * symmetric tridiagonal matrix using the divide and conquer method. 00023 * The eigenvectors of a full or band real symmetric matrix can also be 00024 * found if SSYTRD or SSPTRD or SSBTRD has been used to reduce this 00025 * matrix to tridiagonal form. 00026 * 00027 * This code makes very mild assumptions about floating point 00028 * arithmetic. It will work on machines with a guard digit in 00029 * add/subtract, or on those binary machines without guard digits 00030 * which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. 00031 * It could conceivably fail on hexadecimal or decimal machines 00032 * without guard digits, but we know of none. See SLAED3 for details. 00033 * 00034 * Arguments 00035 * ========= 00036 * 00037 * COMPZ (input) CHARACTER*1 00038 * = 'N': Compute eigenvalues only. 00039 * = 'I': Compute eigenvectors of tridiagonal matrix also. 00040 * = 'V': Compute eigenvectors of original dense symmetric 00041 * matrix also. On entry, Z contains the orthogonal 00042 * matrix used to reduce the original matrix to 00043 * tridiagonal form. 00044 * 00045 * N (input) INTEGER 00046 * The dimension of the symmetric tridiagonal matrix. N >= 0. 00047 * 00048 * D (input/output) REAL array, dimension (N) 00049 * On entry, the diagonal elements of the tridiagonal matrix. 00050 * On exit, if INFO = 0, the eigenvalues in ascending order. 00051 * 00052 * E (input/output) REAL array, dimension (N-1) 00053 * On entry, the subdiagonal elements of the tridiagonal matrix. 00054 * On exit, E has been destroyed. 00055 * 00056 * Z (input/output) REAL array, dimension (LDZ,N) 00057 * On entry, if COMPZ = 'V', then Z contains the orthogonal 00058 * matrix used in the reduction to tridiagonal form. 00059 * On exit, if INFO = 0, then if COMPZ = 'V', Z contains the 00060 * orthonormal eigenvectors of the original symmetric matrix, 00061 * and if COMPZ = 'I', Z contains the orthonormal eigenvectors 00062 * of the symmetric tridiagonal matrix. 00063 * If COMPZ = 'N', then Z is not referenced. 00064 * 00065 * LDZ (input) INTEGER 00066 * The leading dimension of the array Z. LDZ >= 1. 00067 * If eigenvectors are desired, then LDZ >= max(1,N). 00068 * 00069 * WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) 00070 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00071 * 00072 * LWORK (input) INTEGER 00073 * The dimension of the array WORK. 00074 * If COMPZ = 'N' or N <= 1 then LWORK must be at least 1. 00075 * If COMPZ = 'V' and N > 1 then LWORK must be at least 00076 * ( 1 + 3*N + 2*N*lg N + 3*N**2 ), 00077 * where lg( N ) = smallest integer k such 00078 * that 2**k >= N. 00079 * If COMPZ = 'I' and N > 1 then LWORK must be at least 00080 * ( 1 + 4*N + N**2 ). 00081 * Note that for COMPZ = 'I' or 'V', then if N is less than or 00082 * equal to the minimum divide size, usually 25, then LWORK need 00083 * only be max(1,2*(N-1)). 00084 * 00085 * If LWORK = -1, then a workspace query is assumed; the routine 00086 * only calculates the optimal size of the WORK array, returns 00087 * this value as the first entry of the WORK array, and no error 00088 * message related to LWORK is issued by XERBLA. 00089 * 00090 * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) 00091 * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 00092 * 00093 * LIWORK (input) INTEGER 00094 * The dimension of the array IWORK. 00095 * If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1. 00096 * If COMPZ = 'V' and N > 1 then LIWORK must be at least 00097 * ( 6 + 6*N + 5*N*lg N ). 00098 * If COMPZ = 'I' and N > 1 then LIWORK must be at least 00099 * ( 3 + 5*N ). 00100 * Note that for COMPZ = 'I' or 'V', then if N is less than or 00101 * equal to the minimum divide size, usually 25, then LIWORK 00102 * need only be 1. 00103 * 00104 * If LIWORK = -1, then a workspace query is assumed; the 00105 * routine only calculates the optimal size of the IWORK array, 00106 * returns this value as the first entry of the IWORK array, and 00107 * no error message related to LIWORK is issued by XERBLA. 00108 * 00109 * INFO (output) INTEGER 00110 * = 0: successful exit. 00111 * < 0: if INFO = -i, the i-th argument had an illegal value. 00112 * > 0: The algorithm failed to compute an eigenvalue while 00113 * working on the submatrix lying in rows and columns 00114 * INFO/(N+1) through mod(INFO,N+1). 00115 * 00116 * Further Details 00117 * =============== 00118 * 00119 * Based on contributions by 00120 * Jeff Rutter, Computer Science Division, University of California 00121 * at Berkeley, USA 00122 * Modified by Francoise Tisseur, University of Tennessee. 00123 * 00124 * ===================================================================== 00125 * 00126 * .. Parameters .. 00127 REAL ZERO, ONE, TWO 00128 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0 ) 00129 * .. 00130 * .. Local Scalars .. 00131 LOGICAL LQUERY 00132 INTEGER FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN, 00133 $ LWMIN, M, SMLSIZ, START, STOREZ, STRTRW 00134 REAL EPS, ORGNRM, P, TINY 00135 * .. 00136 * .. External Functions .. 00137 LOGICAL LSAME 00138 INTEGER ILAENV 00139 REAL SLAMCH, SLANST 00140 EXTERNAL ILAENV, LSAME, SLAMCH, SLANST 00141 * .. 00142 * .. External Subroutines .. 00143 EXTERNAL SGEMM, SLACPY, SLAED0, SLASCL, SLASET, SLASRT, 00144 $ SSTEQR, SSTERF, SSWAP, XERBLA 00145 * .. 00146 * .. Intrinsic Functions .. 00147 INTRINSIC ABS, INT, LOG, MAX, MOD, REAL, SQRT 00148 * .. 00149 * .. Executable Statements .. 00150 * 00151 * Test the input parameters. 00152 * 00153 INFO = 0 00154 LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) 00155 * 00156 IF( LSAME( COMPZ, 'N' ) ) THEN 00157 ICOMPZ = 0 00158 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN 00159 ICOMPZ = 1 00160 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN 00161 ICOMPZ = 2 00162 ELSE 00163 ICOMPZ = -1 00164 END IF 00165 IF( ICOMPZ.LT.0 ) THEN 00166 INFO = -1 00167 ELSE IF( N.LT.0 ) THEN 00168 INFO = -2 00169 ELSE IF( ( LDZ.LT.1 ) .OR. 00170 $ ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, N ) ) ) THEN 00171 INFO = -6 00172 END IF 00173 * 00174 IF( INFO.EQ.0 ) THEN 00175 * 00176 * Compute the workspace requirements 00177 * 00178 SMLSIZ = ILAENV( 9, 'SSTEDC', ' ', 0, 0, 0, 0 ) 00179 IF( N.LE.1 .OR. ICOMPZ.EQ.0 ) THEN 00180 LIWMIN = 1 00181 LWMIN = 1 00182 ELSE IF( N.LE.SMLSIZ ) THEN 00183 LIWMIN = 1 00184 LWMIN = 2*( N - 1 ) 00185 ELSE 00186 LGN = INT( LOG( REAL( N ) )/LOG( TWO ) ) 00187 IF( 2**LGN.LT.N ) 00188 $ LGN = LGN + 1 00189 IF( 2**LGN.LT.N ) 00190 $ LGN = LGN + 1 00191 IF( ICOMPZ.EQ.1 ) THEN 00192 LWMIN = 1 + 3*N + 2*N*LGN + 3*N**2 00193 LIWMIN = 6 + 6*N + 5*N*LGN 00194 ELSE IF( ICOMPZ.EQ.2 ) THEN 00195 LWMIN = 1 + 4*N + N**2 00196 LIWMIN = 3 + 5*N 00197 END IF 00198 END IF 00199 WORK( 1 ) = LWMIN 00200 IWORK( 1 ) = LIWMIN 00201 * 00202 IF( LWORK.LT.LWMIN .AND. .NOT. LQUERY ) THEN 00203 INFO = -8 00204 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT. LQUERY ) THEN 00205 INFO = -10 00206 END IF 00207 END IF 00208 * 00209 IF( INFO.NE.0 ) THEN 00210 CALL XERBLA( 'SSTEDC', -INFO ) 00211 RETURN 00212 ELSE IF (LQUERY) THEN 00213 RETURN 00214 END IF 00215 * 00216 * Quick return if possible 00217 * 00218 IF( N.EQ.0 ) 00219 $ RETURN 00220 IF( N.EQ.1 ) THEN 00221 IF( ICOMPZ.NE.0 ) 00222 $ Z( 1, 1 ) = ONE 00223 RETURN 00224 END IF 00225 * 00226 * If the following conditional clause is removed, then the routine 00227 * will use the Divide and Conquer routine to compute only the 00228 * eigenvalues, which requires (3N + 3N**2) real workspace and 00229 * (2 + 5N + 2N lg(N)) integer workspace. 00230 * Since on many architectures SSTERF is much faster than any other 00231 * algorithm for finding eigenvalues only, it is used here 00232 * as the default. If the conditional clause is removed, then 00233 * information on the size of workspace needs to be changed. 00234 * 00235 * If COMPZ = 'N', use SSTERF to compute the eigenvalues. 00236 * 00237 IF( ICOMPZ.EQ.0 ) THEN 00238 CALL SSTERF( N, D, E, INFO ) 00239 GO TO 50 00240 END IF 00241 * 00242 * If N is smaller than the minimum divide size (SMLSIZ+1), then 00243 * solve the problem with another solver. 00244 * 00245 IF( N.LE.SMLSIZ ) THEN 00246 * 00247 CALL SSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) 00248 * 00249 ELSE 00250 * 00251 * If COMPZ = 'V', the Z matrix must be stored elsewhere for later 00252 * use. 00253 * 00254 IF( ICOMPZ.EQ.1 ) THEN 00255 STOREZ = 1 + N*N 00256 ELSE 00257 STOREZ = 1 00258 END IF 00259 * 00260 IF( ICOMPZ.EQ.2 ) THEN 00261 CALL SLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) 00262 END IF 00263 * 00264 * Scale. 00265 * 00266 ORGNRM = SLANST( 'M', N, D, E ) 00267 IF( ORGNRM.EQ.ZERO ) 00268 $ GO TO 50 00269 * 00270 EPS = SLAMCH( 'Epsilon' ) 00271 * 00272 START = 1 00273 * 00274 * while ( START <= N ) 00275 * 00276 10 CONTINUE 00277 IF( START.LE.N ) THEN 00278 * 00279 * Let FINISH be the position of the next subdiagonal entry 00280 * such that E( FINISH ) <= TINY or FINISH = N if no such 00281 * subdiagonal exists. The matrix identified by the elements 00282 * between START and FINISH constitutes an independent 00283 * sub-problem. 00284 * 00285 FINISH = START 00286 20 CONTINUE 00287 IF( FINISH.LT.N ) THEN 00288 TINY = EPS*SQRT( ABS( D( FINISH ) ) )* 00289 $ SQRT( ABS( D( FINISH+1 ) ) ) 00290 IF( ABS( E( FINISH ) ).GT.TINY ) THEN 00291 FINISH = FINISH + 1 00292 GO TO 20 00293 END IF 00294 END IF 00295 * 00296 * (Sub) Problem determined. Compute its size and solve it. 00297 * 00298 M = FINISH - START + 1 00299 IF( M.EQ.1 ) THEN 00300 START = FINISH + 1 00301 GO TO 10 00302 END IF 00303 IF( M.GT.SMLSIZ ) THEN 00304 * 00305 * Scale. 00306 * 00307 ORGNRM = SLANST( 'M', M, D( START ), E( START ) ) 00308 CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ), M, 00309 $ INFO ) 00310 CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1, E( START ), 00311 $ M-1, INFO ) 00312 * 00313 IF( ICOMPZ.EQ.1 ) THEN 00314 STRTRW = 1 00315 ELSE 00316 STRTRW = START 00317 END IF 00318 CALL SLAED0( ICOMPZ, N, M, D( START ), E( START ), 00319 $ Z( STRTRW, START ), LDZ, WORK( 1 ), N, 00320 $ WORK( STOREZ ), IWORK, INFO ) 00321 IF( INFO.NE.0 ) THEN 00322 INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) + 00323 $ MOD( INFO, ( M+1 ) ) + START - 1 00324 GO TO 50 00325 END IF 00326 * 00327 * Scale back. 00328 * 00329 CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ), M, 00330 $ INFO ) 00331 * 00332 ELSE 00333 IF( ICOMPZ.EQ.1 ) THEN 00334 * 00335 * Since QR won't update a Z matrix which is larger than 00336 * the length of D, we must solve the sub-problem in a 00337 * workspace and then multiply back into Z. 00338 * 00339 CALL SSTEQR( 'I', M, D( START ), E( START ), WORK, M, 00340 $ WORK( M*M+1 ), INFO ) 00341 CALL SLACPY( 'A', N, M, Z( 1, START ), LDZ, 00342 $ WORK( STOREZ ), N ) 00343 CALL SGEMM( 'N', 'N', N, M, M, ONE, 00344 $ WORK( STOREZ ), N, WORK, M, ZERO, 00345 $ Z( 1, START ), LDZ ) 00346 ELSE IF( ICOMPZ.EQ.2 ) THEN 00347 CALL SSTEQR( 'I', M, D( START ), E( START ), 00348 $ Z( START, START ), LDZ, WORK, INFO ) 00349 ELSE 00350 CALL SSTERF( M, D( START ), E( START ), INFO ) 00351 END IF 00352 IF( INFO.NE.0 ) THEN 00353 INFO = START*( N+1 ) + FINISH 00354 GO TO 50 00355 END IF 00356 END IF 00357 * 00358 START = FINISH + 1 00359 GO TO 10 00360 END IF 00361 * 00362 * endwhile 00363 * 00364 * If the problem split any number of times, then the eigenvalues 00365 * will not be properly ordered. Here we permute the eigenvalues 00366 * (and the associated eigenvectors) into ascending order. 00367 * 00368 IF( M.NE.N ) THEN 00369 IF( ICOMPZ.EQ.0 ) THEN 00370 * 00371 * Use Quick Sort 00372 * 00373 CALL SLASRT( 'I', N, D, INFO ) 00374 * 00375 ELSE 00376 * 00377 * Use Selection Sort to minimize swaps of eigenvectors 00378 * 00379 DO 40 II = 2, N 00380 I = II - 1 00381 K = I 00382 P = D( I ) 00383 DO 30 J = II, N 00384 IF( D( J ).LT.P ) THEN 00385 K = J 00386 P = D( J ) 00387 END IF 00388 30 CONTINUE 00389 IF( K.NE.I ) THEN 00390 D( K ) = D( I ) 00391 D( I ) = P 00392 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 ) 00393 END IF 00394 40 CONTINUE 00395 END IF 00396 END IF 00397 END IF 00398 * 00399 50 CONTINUE 00400 WORK( 1 ) = LWMIN 00401 IWORK( 1 ) = LIWMIN 00402 * 00403 RETURN 00404 * 00405 * End of SSTEDC 00406 * 00407 END
1.7.3 
