1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496
|
subroutine hqror2(nm,n,low,igh,h,wr,wi,z,ierr,job)
c
integer i,j,k,l,m,n,en,ll,mm,na,nm,
x igh,itn,its,low,mp2,enm2,ierr,job
double precision h(nm,n),wr(n),wi(n),z(nm,n)
double precision p,q,r,s,t,w,x,y,zz,norm,tst1,tst2,machep,dlamch
logical notlas
c
c this subroutine is a translation of the algol procedure hqr2,
c num. math. 16, 181-204(1970) by peters and wilkinson.
c handbook for auto. comp., vol.ii-linear algebra, 372-395(1971).
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
c MODIFICATIONS WRT EISPACK VERSION
c ---------------------------------
c 1. 1x1 and 2x2 diagonal blocks are clearly isolated by
c forcing subdiagonal entries to zero
c 2. Merging of hqr/hqr2 driven by a job parameter
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
c This subroutine finds the eigenvalues of a real upper
c hessenberg matrix by the qr method. In addition, the
c orthogonal transformation leading to the Schur form is
c accumulated
c
c on input
c
c nm must be set to the row dimension of two-dimensional
c array parameters as declared in the calling program
c dimension statement.
c
c n is the order of the matrix.
c
c low and igh are integers determined by the balancing
c subroutine balanc. if balanc has not been used,
c set low=1, igh=n.
c
c h contains the upper hessenberg matrix.
c
c z contains the transformation matrix produced by eltran
c after the reduction by elmhes, or by ortran after the
c reduction by orthes, if performed. if the eigenvectors
c of the hessenberg matrix are desired, z must contain the
c identity matrix.
c
c job has the decimal decomposition xy;
c if x=0 hqror2 compute eigen-decomposition of h
c if x=1 hqror2 computes schur decomposition of h
c if x=2 eigenvalues are computed via schur decomposition
c if y=0 coordinate transformation is not accumulated
c if y=1 coordinate transformation is accumulated
c
c
c on output
c
c h contains the Schur form
c
c wr and wi contain the real and imaginary parts,
c respectively, of the eigenvalues. the eigenvalues
c are unordered except that complex conjugate pairs
c of values appear consecutively with the eigenvalue
c having the positive imaginary part first. if an
c error exit is made, the eigenvalues should be correct
c for indices ierr+1,...,n.
c
c z contains the orthogonal transformation to the real schur
c form. If an error exit is made, z may be incorrect.
c
c ierr is set to
c zero for normal return,
c j if the limit of 30*n iterations is exhausted
c while the j-th eigenvalue is being sought.
c
c calls cdiv for complex division.
c
c questions and comments should be directed to burton s. garbow,
c mathematics and computer science div, argonne national laboratory
c
c this version dated august 1983.
c
c ------------------------------------------------------------------
c
jx=job/10
jy=job-10*jx
c
machep=dlamch('p')
c
ierr = 0
norm = 0.0d0
k = 1
c .......... store roots isolated by balanc
c and compute matrix norm ..........
do 50 i = 1, n
c
do 40 j = k, n
40 norm = norm + dabs(h(i,j))
c
k = i
if(jx .eq. 1) goto 50
if (i .ge. low .and. i .le. igh) go to 50
wr(i) = h(i,i)
wi(i) = 0.0d0
50 continue
c
en = igh
t = 0.0d0
itn = 30*n
c .......... search for next eigenvalues ..........
60 if (en .lt. low) go to 340
its = 0
na = en - 1
enm2 = na - 1
c .......... look for single small sub-diagonal element
c for l=en step -1 until low do -- ..........
70 do 80 ll = low, en
l = en + low - ll
if (l .eq. low) go to 100
s = dabs(h(l-1,l-1)) + dabs(h(l,l))
if (s .eq. 0.0d0) s = norm
tst1 = s
tst2 = tst1 + dabs(h(l,l-1))
if (tst2 .eq. tst1) go to 100
80 continue
c .......... form shift ..........
100 x = h(en,en)
if (l .eq. en) go to 270
y = h(na,na)
w = h(en,na) * h(na,en)
if (l .eq. na) go to 280
if (itn .eq. 0) go to 1000
if (its .ne. 10 .and. its .ne. 20) go to 130
c .......... form exceptional shift ..........
t = t + x
c
do 120 i = low, en
120 h(i,i) = h(i,i) - x
c
s = dabs(h(en,na)) + dabs(h(na,enm2))
x = 0.75d0 * s
y = x
w = -0.4375d0 * s * s
130 its = its + 1
itn = itn - 1
c .......... look for two consecutive small
c sub-diagonal elements.
c for m=en-2 step -1 until l do -- ..........
do 140 mm = l, enm2
m = enm2 + l - mm
zz = h(m,m)
r = x - zz
s = y - zz
p = (r * s - w) / h(m+1,m) + h(m,m+1)
q = h(m+1,m+1) - zz - r - s
r = h(m+2,m+1)
s = dabs(p) + dabs(q) + dabs(r)
p = p / s
q = q / s
r = r / s
if (m .eq. l) go to 150
tst1 = dabs(p)*(dabs(h(m-1,m-1)) + dabs(zz) + dabs(h(m+1,m+1)))
tst2 = tst1 + dabs(h(m,m-1))*(dabs(q) + dabs(r))
if (tst2 .eq. tst1) go to 150
140 continue
c
150 mp2 = m + 2
c
do 160 i = mp2, en
h(i,i-2) = 0.0d0
if (i .eq. mp2) go to 160
h(i,i-3) = 0.0d0
160 continue
c .......... double qr step involving rows l to en and
c columns m to en ..........
do 260 k = m, na
notlas = k .ne. na
if (k .eq. m) go to 170
p = h(k,k-1)
q = h(k+1,k-1)
r = 0.0d0
if (notlas) r = h(k+2,k-1)
x = dabs(p) + dabs(q) + dabs(r)
if (x .eq. 0.0d0) go to 260
p = p / x
q = q / x
r = r / x
170 s = dsign(dsqrt(p*p+q*q+r*r),p)
if (k .eq. m) go to 180
h(k,k-1) = -s * x
go to 190
180 if (l .ne. m) h(k,k-1) = -h(k,k-1)
190 p = p + s
x = p / s
y = q / s
zz = r / s
q = q / p
r = r / p
if (notlas) go to 225
c .......... row modification ..........
do 200 j = k, n
p = h(k,j) + q * h(k+1,j)
h(k,j) = h(k,j) - p * x
h(k+1,j) = h(k+1,j) - p * y
200 continue
c
j = min0(en,k+3)
c .......... column modification ..........
do 210 i = 1, j
p = x * h(i,k) + y * h(i,k+1)
h(i,k) = h(i,k) - p
h(i,k+1) = h(i,k+1) - p * q
210 continue
if(jy .eq. 1) then
c .......... accumulate transformations ..........
do 220 i = low, igh
p = x * z(i,k) + y * z(i,k+1)
z(i,k) = z(i,k) - p
z(i,k+1) = z(i,k+1) - p * q
220 continue
endif
go to 255
225 continue
c .......... row modification ..........
do 230 j = k, n
p = h(k,j) + q * h(k+1,j) + r * h(k+2,j)
h(k,j) = h(k,j) - p * x
h(k+1,j) = h(k+1,j) - p * y
h(k+2,j) = h(k+2,j) - p * zz
230 continue
c
j = min0(en,k+3)
c .......... column modification ..........
do 240 i = 1, j
p = x * h(i,k) + y * h(i,k+1) + zz * h(i,k+2)
h(i,k) = h(i,k) - p
h(i,k+1) = h(i,k+1) - p * q
h(i,k+2) = h(i,k+2) - p * r
240 continue
if(jy .eq. 1) then
c .......... accumulate transformations ..........
do 250 i = low, igh
p = x * z(i,k) + y * z(i,k+1) + zz * z(i,k+2)
z(i,k) = z(i,k) - p
z(i,k+1) = z(i,k+1) - p * q
z(i,k+2) = z(i,k+2) - p * r
250 continue
endif
255 continue
c
260 continue
c
go to 70
c .......... one root found ..........
270 h(en,en) = x + t
ccccc ADDED TO MARK BLOCK SEPARATION BY HARD ZEROS
if(en+1.le.n) h(en+1,en)=0.0d0
cccccccccccccccccccccccccccccccccccccccccccccccccc
if (jx.ne.1) then
wr(en) = h(en,en)
wi(en) = 0.0d0
endif
en = na
go to 60
c .......... two roots found ..........
280 p = (y - x) / 2.0d0
q = p * p + w
zz = dsqrt(dabs(q))
h(en,en) = x + t
x = h(en,en)
h(na,na) = y + t
if (q .lt. 0.0d0) go to 320
c .......... real pair ..........
zz = p + dsign(zz,p)
if (jx.ne.1) then
wr(na) = x + zz
wr(en) = wr(na)
if (zz .ne. 0.0d0) wr(en) = x - w / zz
wi(na) = 0.0d0
wi(en) = 0.0d0
endif
x = h(en,na)
s = dabs(x) + dabs(zz)
p = x / s
q = zz / s
r = dsqrt(p*p+q*q)
p = p / r
q = q / r
c .......... row modification ..........
do 290 j = na, n
zz = h(na,j)
h(na,j) = q * zz + p * h(en,j)
h(en,j) = q * h(en,j) - p * zz
290 continue
c .......... column modification ..........
do 300 i = 1, en
zz = h(i,na)
h(i,na) = q * zz + p * h(i,en)
h(i,en) = q * h(i,en) - p * zz
300 continue
if(jy .eq. 1) then
c .......... accumulate transformations ..........
do 310 i = low, igh
zz = z(i,na)
z(i,na) = q * zz + p * z(i,en)
z(i,en) = q * z(i,en) - p * zz
310 continue
endif
ccccc ADDED TO MARK BLOCK SEPARATION BY HARD ZEROS
h(en,na)=0.0d0
if(en+1.le.n) h(en+1,en)=0.0d0
cccccccccccccccccccccccccccccccccccccccccccccccccc
c
go to 330
c .......... complex pair ..........
320 if (jx.ne.1) then
wr(na) = x + p
wr(en) = x + p
wi(na) = zz
wi(en) = -zz
endif
ccccc ADDED TO MARK BLOCK SEPARATION BY HARD ZEROS
if(en+1.le.n) h(en+1,en)=0.0d0
cccccccccccccccccccccccccccccccccccccccccccccccccc
330 en = enm2
go to 60
340 if(jx.ne.0) goto 1001
if (norm .eq. 0.0d+0) go to 1001
c :::::::::: for en=n step -1 until 1 do -- ::::::::::
do 800 nn = 1, n
en = n + 1 - nn
p = wr(en)
q = wi(en)
na = en - 1
q=q+1.0d+0
if (q-1.0d+0) 710, 600, 800
c :::::::::: real vector ::::::::::
600 m = en
h(en,en) = 1.0d+0
if (na .eq. 0) go to 800
c :::::::::: for i=en-1 step -1 until 1 do -- ::::::::::
do 700 ii = 1, na
i = en - ii
w = h(i,i) - p
r = h(i,en)
if (m .gt. na) go to 620
c
do 610 j = m, na
610 r = r + h(i,j) * h(j,en)
c
620 if (wi(i) .ge. 0.0d+0) go to 630
zz = w
s = r
go to 700
630 m = i
if (wi(i) .ne. 0.0d+0) go to 640
t = w
if (w .eq. 0.0d+0) t = machep * norm
h(i,en) = -r / t
go to 700
c :::::::::: solve real equations ::::::::::
640 x = h(i,i+1)
y = h(i+1,i)
q = (wr(i) - p) * (wr(i) - p) + wi(i) * wi(i)
t = (x * s - zz * r) / q
h(i,en) = t
if (abs(x) .le. abs(zz)) go to 650
h(i+1,en) = (-r - w * t) / x
go to 700
650 h(i+1,en) = (-s - y * t) / zz
700 continue
c :::::::::: end real vector ::::::::::
go to 800
c :::::::::: complex vector ::::::::::
710 m = na
c :::::::::: last vector component chosen imaginary so that
c eigenvector matrix is triangular ::::::::::
if (abs(h(en,na)) .le. abs(h(na,en))) go to 720
h(na,na) = q / h(en,na)
h(na,en) = -(h(en,en) - p) / h(en,na)
go to 730
720 z3r=h(na,na)-p
z3=z3r*z3r+q*q
h(na,na)=-h(na,en)*q/z3
h(na,en)=-h(na,en)*z3r/z3
730 h(en,na) = 0.0d+0
h(en,en) = 1.0d+0
enm2 = na - 1
if (enm2 .eq. 0) go to 800
c :::::::::: for i=en-2 step -1 until 1 do -- ::::::::::
do 790 ii = 1, enm2
i = na - ii
w = h(i,i) - p
ra = 0.0d+0
sa = h(i,en)
c
do 760 j = m, na
ra = ra + h(i,j) * h(j,na)
sa = sa + h(i,j) * h(j,en)
760 continue
c
if (wi(i) .ge. 0.0d+0) go to 770
zz = w
r = ra
s = sa
go to 790
770 m = i
if (wi(i) .ne. 0.0d+0) go to 780
z3=w*w+q*q
z3r=-ra*w-sa*q
z3i=ra*q-sa*w
h(i,na)=z3r/z3
h(i,en)=z3i/z3
go to 790
c :::::::::: solve complex equations ::::::::::
780 x = h(i,i+1)
y = h(i+1,i)
vr = (wr(i) - p) * (wr(i) - p) + wi(i) * wi(i) - q * q
vi = (wr(i) - p) * 2.0d+0 * q
if (vr .eq. 0.0d+0 .and. vi .eq. 0.0d+0) vr = machep * norm
x * (abs(w) + abs(q) + abs(x) + abs(y) + abs(zz))
z3r=x*r-zz*ra+q*sa
z3i=x*s-zz*sa-q*ra
z3=vr*vr+vi*vi
h(i,na)=(z3r*vr+z3i*vi)/z3
h(i,en)=(-z3r*vi+z3i*vr)/z3
if (abs(x) .le. abs(zz) + abs(q)) go to 785
h(i+1,na) = (-ra - w * h(i,na) + q * h(i,en)) / x
h(i+1,en) = (-sa - w * h(i,en) - q * h(i,na)) / x
go to 790
785 z3r=-r-y*h(i,na)
z3i=-s-y*h(i,en)
z3=zz*zz+q*q
h(i+1,na)=(z3r*zz+z3i*q)/z3
h(i+1,en)=(-z3r*q+z3i*zz)/z3
790 continue
c :::::::::: end complex vector ::::::::::
800 continue
c :::::::::: end back substitution.
if(jy.eq.0) goto 1001
c vectors of isolated roots ::::::::::
do 840 i = 1, n
if (i .ge. low .and. i .le. igh) go to 840
c
do 820 j = i, n
820 z(i,j) = h(i,j)
c
840 continue
c :::::::::: multiply by transformation matrix to give
c vectors of original full matrix.
c for j=n step -1 until low do -- ::::::::::
do 880 jj = low, n
j = n + low - jj
m = min(j,igh)
c
do 880 i = low, igh
zz = 0.0d+0
c
do 860 k = low, m
860 zz = zz + z(i,k) * h(k,j)
c
z(i,j) = zz
880 continue
c
go to 1001
c .......... set error -- all eigenvalues have not
c converged after 30*n iterations ..........
1000 ierr = en
1001 return
end
subroutine cdiv(ar,ai,br,bi,cr,ci)
double precision ar,ai,br,bi,cr,ci
c
c complex division, (cr,ci) = (ar,ai)/(br,bi)
c
double precision s,ars,ais,brs,bis
s = dabs(br) + dabs(bi)
ars = ar/s
ais = ai/s
brs = br/s
bis = bi/s
s = brs**2 + bis**2
cr = (ars*brs + ais*bis)/s
ci = (ais*brs - ars*bis)/s
return
end
|