00001 /* shseqr.f -- translated by f2c (version 20061008). 00002 You must link the resulting object file with libf2c: 00003 on Microsoft Windows system, link with libf2c.lib; 00004 on Linux or Unix systems, link with .../path/to/libf2c.a -lm 00005 or, if you install libf2c.a in a standard place, with -lf2c -lm 00006 -- in that order, at the end of the command line, as in 00007 cc *.o -lf2c -lm 00008 Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., 00009 00010 http://www.netlib.org/f2c/libf2c.zip 00011 */ 00012 00013 #include "f2c.h" 00014 #include "blaswrap.h" 00015 00016 /* Table of constant values */ 00017 00018 static real c_b11 = 0.f; 00019 static real c_b12 = 1.f; 00020 static integer c__12 = 12; 00021 static integer c__2 = 2; 00022 static integer c__49 = 49; 00023 00024 /* Subroutine */ int shseqr_(char *job, char *compz, integer *n, integer *ilo, 00025 integer *ihi, real *h__, integer *ldh, real *wr, real *wi, real *z__, 00026 integer *ldz, real *work, integer *lwork, integer *info) 00027 { 00028 /* System generated locals */ 00029 address a__1[2]; 00030 integer h_dim1, h_offset, z_dim1, z_offset, i__1, i__2[2], i__3; 00031 real r__1; 00032 char ch__1[2]; 00033 00034 /* Builtin functions */ 00035 /* Subroutine */ int s_cat(char *, char **, integer *, integer *, ftnlen); 00036 00037 /* Local variables */ 00038 integer i__; 00039 real hl[2401] /* was [49][49] */; 00040 integer kbot, nmin; 00041 extern logical lsame_(char *, char *); 00042 logical initz; 00043 real workl[49]; 00044 logical wantt, wantz; 00045 extern /* Subroutine */ int slaqr0_(logical *, logical *, integer *, 00046 integer *, integer *, real *, integer *, real *, real *, integer * 00047 , integer *, real *, integer *, real *, integer *, integer *), 00048 xerbla_(char *, integer *); 00049 extern integer ilaenv_(integer *, char *, char *, integer *, integer *, 00050 integer *, integer *); 00051 extern /* Subroutine */ int slahqr_(logical *, logical *, integer *, 00052 integer *, integer *, real *, integer *, real *, real *, integer * 00053 , integer *, real *, integer *, integer *), slacpy_(char *, 00054 integer *, integer *, real *, integer *, real *, integer *), slaset_(char *, integer *, integer *, real *, real *, 00055 real *, integer *); 00056 logical lquery; 00057 00058 00059 /* -- LAPACK driver routine (version 3.2) -- */ 00060 /* Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.. */ 00061 /* November 2006 */ 00062 00063 /* .. Scalar Arguments .. */ 00064 /* .. */ 00065 /* .. Array Arguments .. */ 00066 /* .. */ 00067 /* Purpose */ 00068 /* ======= */ 00069 00070 /* SHSEQR computes the eigenvalues of a Hessenberg matrix H */ 00071 /* and, optionally, the matrices T and Z from the Schur decomposition */ 00072 /* H = Z T Z**T, where T is an upper quasi-triangular matrix (the */ 00073 /* Schur form), and Z is the orthogonal matrix of Schur vectors. */ 00074 00075 /* Optionally Z may be postmultiplied into an input orthogonal */ 00076 /* matrix Q so that this routine can give the Schur factorization */ 00077 /* of a matrix A which has been reduced to the Hessenberg form H */ 00078 /* by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. */ 00079 00080 /* Arguments */ 00081 /* ========= */ 00082 00083 /* JOB (input) CHARACTER*1 */ 00084 /* = 'E': compute eigenvalues only; */ 00085 /* = 'S': compute eigenvalues and the Schur form T. */ 00086 00087 /* COMPZ (input) CHARACTER*1 */ 00088 /* = 'N': no Schur vectors are computed; */ 00089 /* = 'I': Z is initialized to the unit matrix and the matrix Z */ 00090 /* of Schur vectors of H is returned; */ 00091 /* = 'V': Z must contain an orthogonal matrix Q on entry, and */ 00092 /* the product Q*Z is returned. */ 00093 00094 /* N (input) INTEGER */ 00095 /* The order of the matrix H. N .GE. 0. */ 00096 00097 /* ILO (input) INTEGER */ 00098 /* IHI (input) INTEGER */ 00099 /* It is assumed that H is already upper triangular in rows */ 00100 /* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally */ 00101 /* set by a previous call to SGEBAL, and then passed to SGEHRD */ 00102 /* when the matrix output by SGEBAL is reduced to Hessenberg */ 00103 /* form. Otherwise ILO and IHI should be set to 1 and N */ 00104 /* respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. */ 00105 /* If N = 0, then ILO = 1 and IHI = 0. */ 00106 00107 /* H (input/output) REAL array, dimension (LDH,N) */ 00108 /* On entry, the upper Hessenberg matrix H. */ 00109 /* On exit, if INFO = 0 and JOB = 'S', then H contains the */ 00110 /* upper quasi-triangular matrix T from the Schur decomposition */ 00111 /* (the Schur form); 2-by-2 diagonal blocks (corresponding to */ 00112 /* complex conjugate pairs of eigenvalues) are returned in */ 00113 /* standard form, with H(i,i) = H(i+1,i+1) and */ 00114 /* H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the */ 00115 /* contents of H are unspecified on exit. (The output value of */ 00116 /* H when INFO.GT.0 is given under the description of INFO */ 00117 /* below.) */ 00118 00119 /* Unlike earlier versions of SHSEQR, this subroutine may */ 00120 /* explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 */ 00121 /* or j = IHI+1, IHI+2, ... N. */ 00122 00123 /* LDH (input) INTEGER */ 00124 /* The leading dimension of the array H. LDH .GE. max(1,N). */ 00125 00126 /* WR (output) REAL array, dimension (N) */ 00127 /* WI (output) REAL array, dimension (N) */ 00128 /* The real and imaginary parts, respectively, of the computed */ 00129 /* eigenvalues. If two eigenvalues are computed as a complex */ 00130 /* conjugate pair, they are stored in consecutive elements of */ 00131 /* WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and */ 00132 /* WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in */ 00133 /* the same order as on the diagonal of the Schur form returned */ 00134 /* in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 */ 00135 /* diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and */ 00136 /* WI(i+1) = -WI(i). */ 00137 00138 /* Z (input/output) REAL array, dimension (LDZ,N) */ 00139 /* If COMPZ = 'N', Z is not referenced. */ 00140 /* If COMPZ = 'I', on entry Z need not be set and on exit, */ 00141 /* if INFO = 0, Z contains the orthogonal matrix Z of the Schur */ 00142 /* vectors of H. If COMPZ = 'V', on entry Z must contain an */ 00143 /* N-by-N matrix Q, which is assumed to be equal to the unit */ 00144 /* matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, */ 00145 /* if INFO = 0, Z contains Q*Z. */ 00146 /* Normally Q is the orthogonal matrix generated by SORGHR */ 00147 /* after the call to SGEHRD which formed the Hessenberg matrix */ 00148 /* H. (The output value of Z when INFO.GT.0 is given under */ 00149 /* the description of INFO below.) */ 00150 00151 /* LDZ (input) INTEGER */ 00152 /* The leading dimension of the array Z. if COMPZ = 'I' or */ 00153 /* COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1. */ 00154 00155 /* WORK (workspace/output) REAL array, dimension (LWORK) */ 00156 /* On exit, if INFO = 0, WORK(1) returns an estimate of */ 00157 /* the optimal value for LWORK. */ 00158 00159 /* LWORK (input) INTEGER */ 00160 /* The dimension of the array WORK. LWORK .GE. max(1,N) */ 00161 /* is sufficient and delivers very good and sometimes */ 00162 /* optimal performance. However, LWORK as large as 11*N */ 00163 /* may be required for optimal performance. A workspace */ 00164 /* query is recommended to determine the optimal workspace */ 00165 /* size. */ 00166 00167 /* If LWORK = -1, then SHSEQR does a workspace query. */ 00168 /* In this case, SHSEQR checks the input parameters and */ 00169 /* estimates the optimal workspace size for the given */ 00170 /* values of N, ILO and IHI. The estimate is returned */ 00171 /* in WORK(1). No error message related to LWORK is */ 00172 /* issued by XERBLA. Neither H nor Z are accessed. */ 00173 00174 00175 /* INFO (output) INTEGER */ 00176 /* = 0: successful exit */ 00177 /* .LT. 0: if INFO = -i, the i-th argument had an illegal */ 00178 /* value */ 00179 /* .GT. 0: if INFO = i, SHSEQR failed to compute all of */ 00180 /* the eigenvalues. Elements 1:ilo-1 and i+1:n of WR */ 00181 /* and WI contain those eigenvalues which have been */ 00182 /* successfully computed. (Failures are rare.) */ 00183 00184 /* If INFO .GT. 0 and JOB = 'E', then on exit, the */ 00185 /* remaining unconverged eigenvalues are the eigen- */ 00186 /* values of the upper Hessenberg matrix rows and */ 00187 /* columns ILO through INFO of the final, output */ 00188 /* value of H. */ 00189 00190 /* If INFO .GT. 0 and JOB = 'S', then on exit */ 00191 00192 /* (*) (initial value of H)*U = U*(final value of H) */ 00193 00194 /* where U is an orthogonal matrix. The final */ 00195 /* value of H is upper Hessenberg and quasi-triangular */ 00196 /* in rows and columns INFO+1 through IHI. */ 00197 00198 /* If INFO .GT. 0 and COMPZ = 'V', then on exit */ 00199 00200 /* (final value of Z) = (initial value of Z)*U */ 00201 00202 /* where U is the orthogonal matrix in (*) (regard- */ 00203 /* less of the value of JOB.) */ 00204 00205 /* If INFO .GT. 0 and COMPZ = 'I', then on exit */ 00206 /* (final value of Z) = U */ 00207 /* where U is the orthogonal matrix in (*) (regard- */ 00208 /* less of the value of JOB.) */ 00209 00210 /* If INFO .GT. 0 and COMPZ = 'N', then Z is not */ 00211 /* accessed. */ 00212 00213 /* ================================================================ */ 00214 /* Default values supplied by */ 00215 /* ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK). */ 00216 /* It is suggested that these defaults be adjusted in order */ 00217 /* to attain best performance in each particular */ 00218 /* computational environment. */ 00219 00220 /* ISPEC=12: The SLAHQR vs SLAQR0 crossover point. */ 00221 /* Default: 75. (Must be at least 11.) */ 00222 00223 /* ISPEC=13: Recommended deflation window size. */ 00224 /* This depends on ILO, IHI and NS. NS is the */ 00225 /* number of simultaneous shifts returned */ 00226 /* by ILAENV(ISPEC=15). (See ISPEC=15 below.) */ 00227 /* The default for (IHI-ILO+1).LE.500 is NS. */ 00228 /* The default for (IHI-ILO+1).GT.500 is 3*NS/2. */ 00229 00230 /* ISPEC=14: Nibble crossover point. (See IPARMQ for */ 00231 /* details.) Default: 14% of deflation window */ 00232 /* size. */ 00233 00234 /* ISPEC=15: Number of simultaneous shifts in a multishift */ 00235 /* QR iteration. */ 00236 00237 /* If IHI-ILO+1 is ... */ 00238 00239 /* greater than ...but less ... the */ 00240 /* or equal to ... than default is */ 00241 00242 /* 1 30 NS = 2(+) */ 00243 /* 30 60 NS = 4(+) */ 00244 /* 60 150 NS = 10(+) */ 00245 /* 150 590 NS = ** */ 00246 /* 590 3000 NS = 64 */ 00247 /* 3000 6000 NS = 128 */ 00248 /* 6000 infinity NS = 256 */ 00249 00250 /* (+) By default some or all matrices of this order */ 00251 /* are passed to the implicit double shift routine */ 00252 /* SLAHQR and this parameter is ignored. See */ 00253 /* ISPEC=12 above and comments in IPARMQ for */ 00254 /* details. */ 00255 00256 /* (**) The asterisks (**) indicate an ad-hoc */ 00257 /* function of N increasing from 10 to 64. */ 00258 00259 /* ISPEC=16: Select structured matrix multiply. */ 00260 /* If the number of simultaneous shifts (specified */ 00261 /* by ISPEC=15) is less than 14, then the default */ 00262 /* for ISPEC=16 is 0. Otherwise the default for */ 00263 /* ISPEC=16 is 2. */ 00264 00265 /* ================================================================ */ 00266 /* Based on contributions by */ 00267 /* Karen Braman and Ralph Byers, Department of Mathematics, */ 00268 /* University of Kansas, USA */ 00269 00270 /* ================================================================ */ 00271 /* References: */ 00272 /* K. Braman, R. Byers and R. Mathias, The Multi-Shift QR */ 00273 /* Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 */ 00274 /* Performance, SIAM Journal of Matrix Analysis, volume 23, pages */ 00275 /* 929--947, 2002. */ 00276 00277 /* K. Braman, R. Byers and R. Mathias, The Multi-Shift QR */ 00278 /* Algorithm Part II: Aggressive Early Deflation, SIAM Journal */ 00279 /* of Matrix Analysis, volume 23, pages 948--973, 2002. */ 00280 00281 /* ================================================================ */ 00282 /* .. Parameters .. */ 00283 00284 /* ==== Matrices of order NTINY or smaller must be processed by */ 00285 /* . SLAHQR because of insufficient subdiagonal scratch space. */ 00286 /* . (This is a hard limit.) ==== */ 00287 00288 /* ==== NL allocates some local workspace to help small matrices */ 00289 /* . through a rare SLAHQR failure. NL .GT. NTINY = 11 is */ 00290 /* . required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom- */ 00291 /* . mended. (The default value of NMIN is 75.) Using NL = 49 */ 00292 /* . allows up to six simultaneous shifts and a 16-by-16 */ 00293 /* . deflation window. ==== */ 00294 /* .. */ 00295 /* .. Local Arrays .. */ 00296 /* .. */ 00297 /* .. Local Scalars .. */ 00298 /* .. */ 00299 /* .. External Functions .. */ 00300 /* .. */ 00301 /* .. External Subroutines .. */ 00302 /* .. */ 00303 /* .. Intrinsic Functions .. */ 00304 /* .. */ 00305 /* .. Executable Statements .. */ 00306 00307 /* ==== Decode and check the input parameters. ==== */ 00308 00309 /* Parameter adjustments */ 00310 h_dim1 = *ldh; 00311 h_offset = 1 + h_dim1; 00312 h__ -= h_offset; 00313 --wr; 00314 --wi; 00315 z_dim1 = *ldz; 00316 z_offset = 1 + z_dim1; 00317 z__ -= z_offset; 00318 --work; 00319 00320 /* Function Body */ 00321 wantt = lsame_(job, "S"); 00322 initz = lsame_(compz, "I"); 00323 wantz = initz || lsame_(compz, "V"); 00324 work[1] = (real) max(1,*n); 00325 lquery = *lwork == -1; 00326 00327 *info = 0; 00328 if (! lsame_(job, "E") && ! wantt) { 00329 *info = -1; 00330 } else if (! lsame_(compz, "N") && ! wantz) { 00331 *info = -2; 00332 } else if (*n < 0) { 00333 *info = -3; 00334 } else if (*ilo < 1 || *ilo > max(1,*n)) { 00335 *info = -4; 00336 } else if (*ihi < min(*ilo,*n) || *ihi > *n) { 00337 *info = -5; 00338 } else if (*ldh < max(1,*n)) { 00339 *info = -7; 00340 } else if (*ldz < 1 || wantz && *ldz < max(1,*n)) { 00341 *info = -11; 00342 } else if (*lwork < max(1,*n) && ! lquery) { 00343 *info = -13; 00344 } 00345 00346 if (*info != 0) { 00347 00348 /* ==== Quick return in case of invalid argument. ==== */ 00349 00350 i__1 = -(*info); 00351 xerbla_("SHSEQR", &i__1); 00352 return 0; 00353 00354 } else if (*n == 0) { 00355 00356 /* ==== Quick return in case N = 0; nothing to do. ==== */ 00357 00358 return 0; 00359 00360 } else if (lquery) { 00361 00362 /* ==== Quick return in case of a workspace query ==== */ 00363 00364 slaqr0_(&wantt, &wantz, n, ilo, ihi, &h__[h_offset], ldh, &wr[1], &wi[ 00365 1], ilo, ihi, &z__[z_offset], ldz, &work[1], lwork, info); 00366 /* ==== Ensure reported workspace size is backward-compatible with */ 00367 /* . previous LAPACK versions. ==== */ 00368 /* Computing MAX */ 00369 r__1 = (real) max(1,*n); 00370 work[1] = dmax(r__1,work[1]); 00371 return 0; 00372 00373 } else { 00374 00375 /* ==== copy eigenvalues isolated by SGEBAL ==== */ 00376 00377 i__1 = *ilo - 1; 00378 for (i__ = 1; i__ <= i__1; ++i__) { 00379 wr[i__] = h__[i__ + i__ * h_dim1]; 00380 wi[i__] = 0.f; 00381 /* L10: */ 00382 } 00383 i__1 = *n; 00384 for (i__ = *ihi + 1; i__ <= i__1; ++i__) { 00385 wr[i__] = h__[i__ + i__ * h_dim1]; 00386 wi[i__] = 0.f; 00387 /* L20: */ 00388 } 00389 00390 /* ==== Initialize Z, if requested ==== */ 00391 00392 if (initz) { 00393 slaset_("A", n, n, &c_b11, &c_b12, &z__[z_offset], ldz) 00394 ; 00395 } 00396 00397 /* ==== Quick return if possible ==== */ 00398 00399 if (*ilo == *ihi) { 00400 wr[*ilo] = h__[*ilo + *ilo * h_dim1]; 00401 wi[*ilo] = 0.f; 00402 return 0; 00403 } 00404 00405 /* ==== SLAHQR/SLAQR0 crossover point ==== */ 00406 00407 /* Writing concatenation */ 00408 i__2[0] = 1, a__1[0] = job; 00409 i__2[1] = 1, a__1[1] = compz; 00410 s_cat(ch__1, a__1, i__2, &c__2, (ftnlen)2); 00411 nmin = ilaenv_(&c__12, "SHSEQR", ch__1, n, ilo, ihi, lwork); 00412 nmin = max(11,nmin); 00413 00414 /* ==== SLAQR0 for big matrices; SLAHQR for small ones ==== */ 00415 00416 if (*n > nmin) { 00417 slaqr0_(&wantt, &wantz, n, ilo, ihi, &h__[h_offset], ldh, &wr[1], 00418 &wi[1], ilo, ihi, &z__[z_offset], ldz, &work[1], lwork, 00419 info); 00420 } else { 00421 00422 /* ==== Small matrix ==== */ 00423 00424 slahqr_(&wantt, &wantz, n, ilo, ihi, &h__[h_offset], ldh, &wr[1], 00425 &wi[1], ilo, ihi, &z__[z_offset], ldz, info); 00426 00427 if (*info > 0) { 00428 00429 /* ==== A rare SLAHQR failure! SLAQR0 sometimes succeeds */ 00430 /* . when SLAHQR fails. ==== */ 00431 00432 kbot = *info; 00433 00434 if (*n >= 49) { 00435 00436 /* ==== Larger matrices have enough subdiagonal scratch */ 00437 /* . space to call SLAQR0 directly. ==== */ 00438 00439 slaqr0_(&wantt, &wantz, n, ilo, &kbot, &h__[h_offset], 00440 ldh, &wr[1], &wi[1], ilo, ihi, &z__[z_offset], 00441 ldz, &work[1], lwork, info); 00442 00443 } else { 00444 00445 /* ==== Tiny matrices don't have enough subdiagonal */ 00446 /* . scratch space to benefit from SLAQR0. Hence, */ 00447 /* . tiny matrices must be copied into a larger */ 00448 /* . array before calling SLAQR0. ==== */ 00449 00450 slacpy_("A", n, n, &h__[h_offset], ldh, hl, &c__49); 00451 hl[*n + 1 + *n * 49 - 50] = 0.f; 00452 i__1 = 49 - *n; 00453 slaset_("A", &c__49, &i__1, &c_b11, &c_b11, &hl[(*n + 1) * 00454 49 - 49], &c__49); 00455 slaqr0_(&wantt, &wantz, &c__49, ilo, &kbot, hl, &c__49, & 00456 wr[1], &wi[1], ilo, ihi, &z__[z_offset], ldz, 00457 workl, &c__49, info); 00458 if (wantt || *info != 0) { 00459 slacpy_("A", n, n, hl, &c__49, &h__[h_offset], ldh); 00460 } 00461 } 00462 } 00463 } 00464 00465 /* ==== Clear out the trash, if necessary. ==== */ 00466 00467 if ((wantt || *info != 0) && *n > 2) { 00468 i__1 = *n - 2; 00469 i__3 = *n - 2; 00470 slaset_("L", &i__1, &i__3, &c_b11, &c_b11, &h__[h_dim1 + 3], ldh); 00471 } 00472 00473 /* ==== Ensure reported workspace size is backward-compatible with */ 00474 /* . previous LAPACK versions. ==== */ 00475 00476 /* Computing MAX */ 00477 r__1 = (real) max(1,*n); 00478 work[1] = dmax(r__1,work[1]); 00479 } 00480 00481 /* ==== End of SHSEQR ==== */ 00482 00483 return 0; 00484 } /* shseqr_ */