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subroutine hqr2(nm,n,low,igh,h,wr,wi,z,ierr)
c
integer i,j,k,l,m,n,en,ii,jj,ll,mm,na,nm,nn,
x igh,itn,its,low,mp2,enm2,ierr
double precision h(nm,n),wr(n),wi(n),z(nm,n)
double precision p,q,r,s,t,w,x,y,ra,sa,vi,vr,zz,norm,tst1,tst2
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
c this subroutine finds the eigenvalues and eigenvectors
c of a real upper hessenberg matrix by the qr method. the
c eigenvectors of a real general matrix can also be found
c if elmhes and eltran or orthes and ortran have
c been used to reduce this general matrix to hessenberg form
c and to accumulate the similarity transformations.
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 on output
c
c h has been destroyed.
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 real and imaginary parts of the eigenvectors.
c if the i-th eigenvalue is real, the i-th column of z
c contains its eigenvector. if the i-th eigenvalue is complex
c with positive imaginary part, the i-th and (i+1)-th
c columns of z contain the real and imaginary parts of its
c eigenvector. the eigenvectors are unnormalized. if an
c error exit is made, none of the eigenvectors has been found.
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
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 (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
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
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
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
255 continue
c
260 continue
c
go to 70
c .......... one root found ..........
270 h(en,en) = x + t
wr(en) = h(en,en)
wi(en) = 0.0d0
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)
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
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
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
c
go to 330
c .......... complex pair ..........
320 wr(na) = x + p
wr(en) = x + p
wi(na) = zz
wi(en) = -zz
330 en = enm2
go to 60
c .......... all roots found. backsubstitute to find
c vectors of upper triangular form ..........
340 if (norm .eq. 0.0d0) 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
if (q) 710, 600, 800
c .......... real vector ..........
600 m = en
h(en,en) = 1.0d0
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 = 0.0d0
c
do 610 j = m, en
610 r = r + h(i,j) * h(j,en)
c
if (wi(i) .ge. 0.0d0) go to 630
zz = w
s = r
go to 700
630 m = i
if (wi(i) .ne. 0.0d0) go to 640
t = w
if (t .ne. 0.0d0) go to 635
tst1 = norm
t = tst1
632 t = 0.01d0 * t
tst2 = norm + t
if (tst2 .gt. tst1) go to 632
635 h(i,en) = -r / t
go to 680
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 (dabs(x) .le. dabs(zz)) go to 650
h(i+1,en) = (-r - w * t) / x
go to 680
650 h(i+1,en) = (-s - y * t) / zz
c
c .......... overflow control ..........
680 t = dabs(h(i,en))
if (t .eq. 0.0d0) go to 700
tst1 = t
tst2 = tst1 + 1.0d0/tst1
if (tst2 .gt. tst1) go to 700
do 690 j = i, en
h(j,en) = h(j,en)/t
690 continue
c
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 (dabs(h(en,na)) .le. dabs(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 call cdiv(0.0d0,-h(na,en),h(na,na)-p,q,h(na,na),h(na,en))
730 h(en,na) = 0.0d0
h(en,en) = 1.0d0
enm2 = na - 1
if (enm2 .eq. 0) go to 800
c .......... for i=en-2 step -1 until 1 do -- ..........
do 795 ii = 1, enm2
i = na - ii
w = h(i,i) - p
ra = 0.0d0
sa = 0.0d0
c
do 760 j = m, en
ra = ra + h(i,j) * h(j,na)
sa = sa + h(i,j) * h(j,en)
760 continue
c
if (wi(i) .ge. 0.0d0) go to 770
zz = w
r = ra
s = sa
go to 795
770 m = i
if (wi(i) .ne. 0.0d0) go to 780
call cdiv(-ra,-sa,w,q,h(i,na),h(i,en))
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) * 2.0d0 * q
if (vr .ne. 0.0d0 .or. vi .ne. 0.0d0) go to 784
tst1 = norm * (dabs(w) + dabs(q) + dabs(x)
x + dabs(y) + dabs(zz))
vr = tst1
783 vr = 0.01d0 * vr
tst2 = tst1 + vr
if (tst2 .gt. tst1) go to 783
784 call cdiv(x*r-zz*ra+q*sa,x*s-zz*sa-q*ra,vr,vi,
x h(i,na),h(i,en))
if (dabs(x) .le. dabs(zz) + dabs(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 call cdiv(-r-y*h(i,na),-s-y*h(i,en),zz,q,
x h(i+1,na),h(i+1,en))
c
c .......... overflow control ..........
790 t = dmax1(dabs(h(i,na)), dabs(h(i,en)))
if (t .eq. 0.0d0) go to 795
tst1 = t
tst2 = tst1 + 1.0d0/tst1
if (tst2 .gt. tst1) go to 795
do 792 j = i, en
h(j,na) = h(j,na)/t
h(j,en) = h(j,en)/t
792 continue
c
795 continue
c .......... end complex vector ..........
800 continue
c .......... end back substitution.
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 = min0(j,igh)
c
do 880 i = low, igh
zz = 0.0d0
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
|
