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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 SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, 00002 $ LDZ, WORK, LWORK, INFO ) 00003 * 00004 * -- LAPACK computational routine (version 3.2.2) -- 00005 * Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.. 00006 * June 2010 00007 * 00008 * .. Scalar Arguments .. 00009 INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N 00010 CHARACTER COMPZ, JOB 00011 * .. 00012 * .. Array Arguments .. 00013 REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ), 00014 $ Z( LDZ, * ) 00015 * .. 00016 * Purpose 00017 * ======= 00018 * 00019 * SHSEQR computes the eigenvalues of a Hessenberg matrix H 00020 * and, optionally, the matrices T and Z from the Schur decomposition 00021 * H = Z T Z**T, where T is an upper quasi-triangular matrix (the 00022 * Schur form), and Z is the orthogonal matrix of Schur vectors. 00023 * 00024 * Optionally Z may be postmultiplied into an input orthogonal 00025 * matrix Q so that this routine can give the Schur factorization 00026 * of a matrix A which has been reduced to the Hessenberg form H 00027 * by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. 00028 * 00029 * Arguments 00030 * ========= 00031 * 00032 * JOB (input) CHARACTER*1 00033 * = 'E': compute eigenvalues only; 00034 * = 'S': compute eigenvalues and the Schur form T. 00035 * 00036 * COMPZ (input) CHARACTER*1 00037 * = 'N': no Schur vectors are computed; 00038 * = 'I': Z is initialized to the unit matrix and the matrix Z 00039 * of Schur vectors of H is returned; 00040 * = 'V': Z must contain an orthogonal matrix Q on entry, and 00041 * the product Q*Z is returned. 00042 * 00043 * N (input) INTEGER 00044 * The order of the matrix H. N .GE. 0. 00045 * 00046 * ILO (input) INTEGER 00047 * IHI (input) INTEGER 00048 * It is assumed that H is already upper triangular in rows 00049 * and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally 00050 * set by a previous call to SGEBAL, and then passed to SGEHRD 00051 * when the matrix output by SGEBAL is reduced to Hessenberg 00052 * form. Otherwise ILO and IHI should be set to 1 and N 00053 * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. 00054 * If N = 0, then ILO = 1 and IHI = 0. 00055 * 00056 * H (input/output) REAL array, dimension (LDH,N) 00057 * On entry, the upper Hessenberg matrix H. 00058 * On exit, if INFO = 0 and JOB = 'S', then H contains the 00059 * upper quasi-triangular matrix T from the Schur decomposition 00060 * (the Schur form); 2-by-2 diagonal blocks (corresponding to 00061 * complex conjugate pairs of eigenvalues) are returned in 00062 * standard form, with H(i,i) = H(i+1,i+1) and 00063 * H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the 00064 * contents of H are unspecified on exit. (The output value of 00065 * H when INFO.GT.0 is given under the description of INFO 00066 * below.) 00067 * 00068 * Unlike earlier versions of SHSEQR, this subroutine may 00069 * explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 00070 * or j = IHI+1, IHI+2, ... N. 00071 * 00072 * LDH (input) INTEGER 00073 * The leading dimension of the array H. LDH .GE. max(1,N). 00074 * 00075 * WR (output) REAL array, dimension (N) 00076 * WI (output) REAL array, dimension (N) 00077 * The real and imaginary parts, respectively, of the computed 00078 * eigenvalues. If two eigenvalues are computed as a complex 00079 * conjugate pair, they are stored in consecutive elements of 00080 * WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and 00081 * WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in 00082 * the same order as on the diagonal of the Schur form returned 00083 * in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 00084 * diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and 00085 * WI(i+1) = -WI(i). 00086 * 00087 * Z (input/output) REAL array, dimension (LDZ,N) 00088 * If COMPZ = 'N', Z is not referenced. 00089 * If COMPZ = 'I', on entry Z need not be set and on exit, 00090 * if INFO = 0, Z contains the orthogonal matrix Z of the Schur 00091 * vectors of H. If COMPZ = 'V', on entry Z must contain an 00092 * N-by-N matrix Q, which is assumed to be equal to the unit 00093 * matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, 00094 * if INFO = 0, Z contains Q*Z. 00095 * Normally Q is the orthogonal matrix generated by SORGHR 00096 * after the call to SGEHRD which formed the Hessenberg matrix 00097 * H. (The output value of Z when INFO.GT.0 is given under 00098 * the description of INFO below.) 00099 * 00100 * LDZ (input) INTEGER 00101 * The leading dimension of the array Z. if COMPZ = 'I' or 00102 * COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1. 00103 * 00104 * WORK (workspace/output) REAL array, dimension (LWORK) 00105 * On exit, if INFO = 0, WORK(1) returns an estimate of 00106 * the optimal value for LWORK. 00107 * 00108 * LWORK (input) INTEGER 00109 * The dimension of the array WORK. LWORK .GE. max(1,N) 00110 * is sufficient and delivers very good and sometimes 00111 * optimal performance. However, LWORK as large as 11*N 00112 * may be required for optimal performance. A workspace 00113 * query is recommended to determine the optimal workspace 00114 * size. 00115 * 00116 * If LWORK = -1, then SHSEQR does a workspace query. 00117 * In this case, SHSEQR checks the input parameters and 00118 * estimates the optimal workspace size for the given 00119 * values of N, ILO and IHI. The estimate is returned 00120 * in WORK(1). No error message related to LWORK is 00121 * issued by XERBLA. Neither H nor Z are accessed. 00122 * 00123 * 00124 * INFO (output) INTEGER 00125 * = 0: successful exit 00126 * .LT. 0: if INFO = -i, the i-th argument had an illegal 00127 * value 00128 * .GT. 0: if INFO = i, SHSEQR failed to compute all of 00129 * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR 00130 * and WI contain those eigenvalues which have been 00131 * successfully computed. (Failures are rare.) 00132 * 00133 * If INFO .GT. 0 and JOB = 'E', then on exit, the 00134 * remaining unconverged eigenvalues are the eigen- 00135 * values of the upper Hessenberg matrix rows and 00136 * columns ILO through INFO of the final, output 00137 * value of H. 00138 * 00139 * If INFO .GT. 0 and JOB = 'S', then on exit 00140 * 00141 * (*) (initial value of H)*U = U*(final value of H) 00142 * 00143 * where U is an orthogonal matrix. The final 00144 * value of H is upper Hessenberg and quasi-triangular 00145 * in rows and columns INFO+1 through IHI. 00146 * 00147 * If INFO .GT. 0 and COMPZ = 'V', then on exit 00148 * 00149 * (final value of Z) = (initial value of Z)*U 00150 * 00151 * where U is the orthogonal matrix in (*) (regard- 00152 * less of the value of JOB.) 00153 * 00154 * If INFO .GT. 0 and COMPZ = 'I', then on exit 00155 * (final value of Z) = U 00156 * where U is the orthogonal matrix in (*) (regard- 00157 * less of the value of JOB.) 00158 * 00159 * If INFO .GT. 0 and COMPZ = 'N', then Z is not 00160 * accessed. 00161 * 00162 * ================================================================ 00163 * Default values supplied by 00164 * ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK). 00165 * It is suggested that these defaults be adjusted in order 00166 * to attain best performance in each particular 00167 * computational environment. 00168 * 00169 * ISPEC=12: The SLAHQR vs SLAQR0 crossover point. 00170 * Default: 75. (Must be at least 11.) 00171 * 00172 * ISPEC=13: Recommended deflation window size. 00173 * This depends on ILO, IHI and NS. NS is the 00174 * number of simultaneous shifts returned 00175 * by ILAENV(ISPEC=15). (See ISPEC=15 below.) 00176 * The default for (IHI-ILO+1).LE.500 is NS. 00177 * The default for (IHI-ILO+1).GT.500 is 3*NS/2. 00178 * 00179 * ISPEC=14: Nibble crossover point. (See IPARMQ for 00180 * details.) Default: 14% of deflation window 00181 * size. 00182 * 00183 * ISPEC=15: Number of simultaneous shifts in a multishift 00184 * QR iteration. 00185 * 00186 * If IHI-ILO+1 is ... 00187 * 00188 * greater than ...but less ... the 00189 * or equal to ... than default is 00190 * 00191 * 1 30 NS = 2(+) 00192 * 30 60 NS = 4(+) 00193 * 60 150 NS = 10(+) 00194 * 150 590 NS = ** 00195 * 590 3000 NS = 64 00196 * 3000 6000 NS = 128 00197 * 6000 infinity NS = 256 00198 * 00199 * (+) By default some or all matrices of this order 00200 * are passed to the implicit double shift routine 00201 * SLAHQR and this parameter is ignored. See 00202 * ISPEC=12 above and comments in IPARMQ for 00203 * details. 00204 * 00205 * (**) The asterisks (**) indicate an ad-hoc 00206 * function of N increasing from 10 to 64. 00207 * 00208 * ISPEC=16: Select structured matrix multiply. 00209 * If the number of simultaneous shifts (specified 00210 * by ISPEC=15) is less than 14, then the default 00211 * for ISPEC=16 is 0. Otherwise the default for 00212 * ISPEC=16 is 2. 00213 * 00214 * ================================================================ 00215 * Based on contributions by 00216 * Karen Braman and Ralph Byers, Department of Mathematics, 00217 * University of Kansas, USA 00218 * 00219 * ================================================================ 00220 * References: 00221 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 00222 * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 00223 * Performance, SIAM Journal of Matrix Analysis, volume 23, pages 00224 * 929--947, 2002. 00225 * 00226 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 00227 * Algorithm Part II: Aggressive Early Deflation, SIAM Journal 00228 * of Matrix Analysis, volume 23, pages 948--973, 2002. 00229 * 00230 * ================================================================ 00231 * .. Parameters .. 00232 * 00233 * ==== Matrices of order NTINY or smaller must be processed by 00234 * . SLAHQR because of insufficient subdiagonal scratch space. 00235 * . (This is a hard limit.) ==== 00236 INTEGER NTINY 00237 PARAMETER ( NTINY = 11 ) 00238 * 00239 * ==== NL allocates some local workspace to help small matrices 00240 * . through a rare SLAHQR failure. NL .GT. NTINY = 11 is 00241 * . required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom- 00242 * . mended. (The default value of NMIN is 75.) Using NL = 49 00243 * . allows up to six simultaneous shifts and a 16-by-16 00244 * . deflation window. ==== 00245 INTEGER NL 00246 PARAMETER ( NL = 49 ) 00247 REAL ZERO, ONE 00248 PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0 ) 00249 * .. 00250 * .. Local Arrays .. 00251 REAL HL( NL, NL ), WORKL( NL ) 00252 * .. 00253 * .. Local Scalars .. 00254 INTEGER I, KBOT, NMIN 00255 LOGICAL INITZ, LQUERY, WANTT, WANTZ 00256 * .. 00257 * .. External Functions .. 00258 INTEGER ILAENV 00259 LOGICAL LSAME 00260 EXTERNAL ILAENV, LSAME 00261 * .. 00262 * .. External Subroutines .. 00263 EXTERNAL SLACPY, SLAHQR, SLAQR0, SLASET, XERBLA 00264 * .. 00265 * .. Intrinsic Functions .. 00266 INTRINSIC MAX, MIN, REAL 00267 * .. 00268 * .. Executable Statements .. 00269 * 00270 * ==== Decode and check the input parameters. ==== 00271 * 00272 WANTT = LSAME( JOB, 'S' ) 00273 INITZ = LSAME( COMPZ, 'I' ) 00274 WANTZ = INITZ .OR. LSAME( COMPZ, 'V' ) 00275 WORK( 1 ) = REAL( MAX( 1, N ) ) 00276 LQUERY = LWORK.EQ.-1 00277 * 00278 INFO = 0 00279 IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN 00280 INFO = -1 00281 ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN 00282 INFO = -2 00283 ELSE IF( N.LT.0 ) THEN 00284 INFO = -3 00285 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN 00286 INFO = -4 00287 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN 00288 INFO = -5 00289 ELSE IF( LDH.LT.MAX( 1, N ) ) THEN 00290 INFO = -7 00291 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN 00292 INFO = -11 00293 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN 00294 INFO = -13 00295 END IF 00296 * 00297 IF( INFO.NE.0 ) THEN 00298 * 00299 * ==== Quick return in case of invalid argument. ==== 00300 * 00301 CALL XERBLA( 'SHSEQR', -INFO ) 00302 RETURN 00303 * 00304 ELSE IF( N.EQ.0 ) THEN 00305 * 00306 * ==== Quick return in case N = 0; nothing to do. ==== 00307 * 00308 RETURN 00309 * 00310 ELSE IF( LQUERY ) THEN 00311 * 00312 * ==== Quick return in case of a workspace query ==== 00313 * 00314 CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO, 00315 $ IHI, Z, LDZ, WORK, LWORK, INFO ) 00316 * ==== Ensure reported workspace size is backward-compatible with 00317 * . previous LAPACK versions. ==== 00318 WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) ) 00319 RETURN 00320 * 00321 ELSE 00322 * 00323 * ==== copy eigenvalues isolated by SGEBAL ==== 00324 * 00325 DO 10 I = 1, ILO - 1 00326 WR( I ) = H( I, I ) 00327 WI( I ) = ZERO 00328 10 CONTINUE 00329 DO 20 I = IHI + 1, N 00330 WR( I ) = H( I, I ) 00331 WI( I ) = ZERO 00332 20 CONTINUE 00333 * 00334 * ==== Initialize Z, if requested ==== 00335 * 00336 IF( INITZ ) 00337 $ CALL SLASET( 'A', N, N, ZERO, ONE, Z, LDZ ) 00338 * 00339 * ==== Quick return if possible ==== 00340 * 00341 IF( ILO.EQ.IHI ) THEN 00342 WR( ILO ) = H( ILO, ILO ) 00343 WI( ILO ) = ZERO 00344 RETURN 00345 END IF 00346 * 00347 * ==== SLAHQR/SLAQR0 crossover point ==== 00348 * 00349 NMIN = ILAENV( 12, 'SHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N, 00350 $ ILO, IHI, LWORK ) 00351 NMIN = MAX( NTINY, NMIN ) 00352 * 00353 * ==== SLAQR0 for big matrices; SLAHQR for small ones ==== 00354 * 00355 IF( N.GT.NMIN ) THEN 00356 CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO, 00357 $ IHI, Z, LDZ, WORK, LWORK, INFO ) 00358 ELSE 00359 * 00360 * ==== Small matrix ==== 00361 * 00362 CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO, 00363 $ IHI, Z, LDZ, INFO ) 00364 * 00365 IF( INFO.GT.0 ) THEN 00366 * 00367 * ==== A rare SLAHQR failure! SLAQR0 sometimes succeeds 00368 * . when SLAHQR fails. ==== 00369 * 00370 KBOT = INFO 00371 * 00372 IF( N.GE.NL ) THEN 00373 * 00374 * ==== Larger matrices have enough subdiagonal scratch 00375 * . space to call SLAQR0 directly. ==== 00376 * 00377 CALL SLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, WR, 00378 $ WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO ) 00379 * 00380 ELSE 00381 * 00382 * ==== Tiny matrices don't have enough subdiagonal 00383 * . scratch space to benefit from SLAQR0. Hence, 00384 * . tiny matrices must be copied into a larger 00385 * . array before calling SLAQR0. ==== 00386 * 00387 CALL SLACPY( 'A', N, N, H, LDH, HL, NL ) 00388 HL( N+1, N ) = ZERO 00389 CALL SLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ), 00390 $ NL ) 00391 CALL SLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, WR, 00392 $ WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO ) 00393 IF( WANTT .OR. INFO.NE.0 ) 00394 $ CALL SLACPY( 'A', N, N, HL, NL, H, LDH ) 00395 END IF 00396 END IF 00397 END IF 00398 * 00399 * ==== Clear out the trash, if necessary. ==== 00400 * 00401 IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 ) 00402 $ CALL SLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH ) 00403 * 00404 * ==== Ensure reported workspace size is backward-compatible with 00405 * . previous LAPACK versions. ==== 00406 * 00407 WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) ) 00408 END IF 00409 * 00410 * ==== End of SHSEQR ==== 00411 * 00412 END
1.7.3 
