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subroutine syhsc(n,m,a,na,b,mb,c,z,eps,wrk1,nwrk1,wrk2,nwrk2,iwrk,
& niwrk,ierr)
C
C! calling sequence
C subroutine syhsc(n,m,a,na,b,mb,c,z,wrk1,nwrk1,wrk2,nwrk2,
C + iwrk,niwrk,ierr)
C
C integer n,m,na,mb,nwrk1,nwrk2,niwrk,ierr
C integer iwrk(niwrk)
C double precision a(na,n),b(mb,m),c(na,m)
C double precision z(mb,m),wrk1(nwrk1),wrk2(nwrk2)
C
C arguments in
C
C n integer
C -order of the matrix a.
C
C m integer
C -order of the matrix b.
C
C a double precision(n,n)
C -the coefficient matrix a of the equation.
C n.b. a is overwritten by this routine.
C
C na integer
C -the declared first dimension of a and c .
C
C b double precision(m,m)
C -the coefficient matrix b of the equation.
C n.b. b is overwritten by this routine.
C
C mb integer
C -the declared first dimension of b .
C
C c double precision(n,m)
C -the coefficient matrix c of the equation.
C n.b. c is overwritten by this routine.
C
C nwrk1 integer
C -the length of the internal vector wrk1
C nwrk1 must be at least 2*n**2 + 7*n
C
C nwrk2 integer
C -the length of the internal vector wrk2
C nwrk2 must be at least max(n,m)
C
C niwrk integer
C -the length of the internal vector iwrk
C niwrk must be at least 4*n
C
C arguments out
C
C c double precision(n,m)
C -on return, the solution matrix, x is contained in c
C
C z double precision(m,m)
C -on return, z contains the orthogonal matrix used
C to transform transpose(b) to real upper schur form.
C
C ierr integer
C -error indicator
C
C ierr = 0 successful return
C
C ierr in (1,m ierr-th eigenvalue of b has not been
C determined after 30 iterations.
C
C ierr > m a singular matrix was encountered whilst
C solving for the (ierr-m)th column of x
C
C!working space
C
C wrk1 double precision(nwrk1)
C -where nwrk1 .ge. 2*n**2 + 7*n
C
C wrk2 double precision(nwrk2)
C -where nwrk2 .ge. max(n,m)
C
C iwrk integer(niwrk)
C -where niwrk .ge. 4*n
C
C!purpose
C
C to solve the continuous-time sylvester equation
C ax + xb = c
C
C!method
C
C a is transformed to upper hessenberg form, and the transpose
C of b is transformed to real upper schur form using orthogonal
C transformations. the matrix c is also multiplied by these
C transformation matrices, and the solution of this transformed
C system is computed. this solution is then multiplied by the
C transformation matrices in order to obtain the solution to
C the original problem.
C
C!reference
C
C g.golub, s.nash, and c.f.vanloan," a hessenberg-schur method
C for the problem ax + xb = c ",ieee trans. auto. control,
C vol. ac-24, no. 6, pp. 909-912 (1979).
C
C!auxiliary routines
C
C orthes,ortran (eispack)
C hqror2,transf,nsolve,hesolv,backsb,n2solv,h2solv,backs2
C irow1,irow2,lrow2
C
C!origin:
C
C g.golub,s.nash,c.van loan, dept.comp.sci.,stanford
C university,california january 1982
C
C!
C********************************************************************
C
integer n,m,na,mb,nwrk1,nwrk2,niwrk,ierr
integer iwrk(niwrk)
double precision a(na,n),b(mb,m),c(na,m),z(mb,m)
double precision wrk1(nwrk1),wrk2(nwrk2)
C
C
double precision eps,t(1),swop,reps
do 5 i = 1,m
do 5 j = i,m
swop = b(i,j)
b(i,j) = b(j,i)
b(j,i) = swop
5 continue
C
call orthes(mb,m,1,m,b,wrk2)
call ortran(mb,m,1,m,b,wrk2,z)
call hqror2(mb,m,1,m,b,t,t,z,ierr,11)
C
call orthes(na,n,1,n,a,wrk2)
C
call transf(a,wrk2,1,c,z,0,n,m,na,mb,wrk1,nwrk1)
C
reps = eps * m * m * n * n
ind = m - 1
if (ind .eq. 0) goto 40
10 if (abs(b(ind+1,ind)) .le. reps) goto 20
C
call n2solv(a,b,c,wrk1,nwrk1,mb,m,na,n,ind,iwrk,niwrk,reps,ierr)
C
if (ierr .ne. 0) return
goto 30
C
20 call nsolve(a,b,c,wrk1,nwrk1,mb,m,na,n,ind,iwrk,niwrk,reps,ierr)
C
if (ierr .ne. 0) return
30 if (ind .gt. 0) goto 10
C
40 if (ind .eq. 0)
& call nsolve(a,b,c,wrk1,nwrk1,mb,m,na,n,ind,iwrk,niwrk,reps,ierr)
C
call transf(a,wrk2,0,c,z,1,n,m,na,mb,wrk1,nwrk1)
C
return
end
subroutine transf(a,ort,it1,c,v,it2,m,n,mdim,ndim,d,nwrk1)
c!
integer i,it1,it2,j,k,k1,k2,kk,m,mdim,n,nwrk1
double precision v(ndim,n),c(mdim,n),a(mdim,m),ort(m),d(nwrk1)
double precision g
m2=m-2
if(m2.le.0) go to 45
do 40 kk=1,m2
k=m2-kk+1
if(it1.eq.1) k=kk
k1=k+1
if(a(k1,k).eq.0.0d+0) go to 40
d(k1)=ort(k1)
k2=k+2
do 10 i=k2,m
d(i)=a(i,k)
10 continue
do 30 j=1,n
g=0.0d+0
do 20 i=k1,m
g=g+d(i)*c(i,j)
20 continue
g=g/ort(k1)/a(k1,k)
do 30 i=k1,m
c(i,j)=c(i,j)+g*d(i)
30 continue
40 continue
45 do 60 i=1,m
do 50 j=1,n
d(j)=0.0d+0
do 50 k=1,n
if(it2.eq.0) d(j)=d(j)+c(i,k)*v(k,j)
if(it2.eq.1) d(j)=d(j)+c(i,k)*v(j,k)
50 continue
do 60 j=1,n
c(i,j)=d(j)
60 continue
return
end
subroutine nsolve(a,b,c,d,nwrk1,ndim,n,mdim,m,ind,ipr,niwrk,
+ reps,ierr)
c% calling sequence
c subroutine nsolve(a,b,c,d,nwrk1,ndim,n,mdim,m,ind,ipr,niwrk,
c + reps,ierr)
c integer nwrk1,niwrk
c integer i,i1,ierr,ind,ipr(niwrk),irow1,j,m,m1,mdim,n,ndim,mfin
c double precision a(mdim,m),b(ndim,n),c(mdim,n),d(nwrk1),reps
c% purpose
c this routine is only to be called from syhsc
c%
integer nwrk1,niwrk
integer i,i1,ierr,ind,ipr(niwrk),irow1,j,m,m1,mdim,n,ndim,mfin
double precision a(mdim,m),b(ndim,n),c(mdim,n),d(nwrk1),reps
c
if(ind.lt.n-1) call backsb(c,b,ind,n,m,mdim,ndim)
c
mfin=(m*(m+1))/2+m
do 20 i=1,m
m1=irow1(i,m)
i1=i-1
if(i.eq.1)i1=1
do 10 j=i1,m
ip = m1+j - i1 + 1
d(ip)=a(i,j)
10 continue
j=m1+2
if(i.eq.1)j=j-1
d(j)=d(j)+b(ind+1,ind+1)
ip = mfin + i
d(ip)=c(i,ind+1)
20 continue
c
call hesolv(d,nwrk1,ipr,niwrk,m,reps,ierr)
c
if(ierr.ne.0) go to 40
do 30 i=1,m
ip = ipr(i)
c(i,ind+1)=d(ip)
30 continue
ind=ind-1
return
40 ierr=n+ind-1
return
end
subroutine hesolv(d,nwrk1,ipr,niwrk,m,reps,ierr)
integer i,i1,i2,ierr,ipr(niwrk),irow1,j,j1,k,mfin
double precision d(nwrk1),mult,reps
ierr=0
mfin=(m*(m+1))/2+m
do 10 i=1,m
ip = m + i
ipr(ip)=irow1(i,m)
ipr(i)=i+mfin
10 continue
m1=m-1
if(m.eq.1) go to 35
do 30 i=1,m1
ip = m + i
ipl = ipr(ip)
ipi = ipr(ip+1)
if(abs(d(ipl+1)).gt.abs(d(ipi+1)))go to 20
ipr(ip)=ipr(ip+1)
ipr(ip+1)=ipl
k=ipr(i)
ipr(i)=ipr(i+1)
ipr(i+1)=k
20 if(abs(d(ipr(m+i)+1)).lt.reps) go to 60
ipr(m+i+1)=ipr(m+i+1)+1
mult=d(ipr(m+i+1))/d(ipr(m+i)+1)
d(ipr(i+1))=d(ipr(i+1))-mult*d(ipr(i))
i1=i+1
do 30 j=i1,m
d(ipr(m+i+1)+j-i)=d(ipr(m+i+1)+j-i)-
* mult*d(ipr(m+i)+j+1-i)
30 continue
35 if(abs(d(ipr(m+m)+1)).lt.reps) go to 60
d(ipr(m))=d(ipr(m))/d(ipr(m+m)+1)
if(m1.eq.0) return
do 50 i1=1,m1
i=m-i1
i2=i+1
mult=0.0d+0
do 40 j1=i2,m
j=j1-i2+2
mult=mult+d(ipr(j1))*d(ipr(m+i)+j)
40 continue
d(ipr(i))=(d(ipr(i))-mult)/d(ipr(m+i)+1)
50 continue
return
60 ierr=-1
return
end
subroutine backsb(c,b,ind,n,m,mdim,ndim)
integer i,ind,ind1,ind2,j,m,mdim,n,ndim
double precision b(ndim,n),c(mdim,n)
ind1=ind+1
ind2=ind+2
do 10 i=ind2,n
do 10 j=1,m
c(j,ind1)=c(j,ind1)-b(ind1,i)*c(j,i)
10 continue
return
end
subroutine n2solv(a,b,c,d,nwrk1,ndim,n,mdim,m,ind,ipr,niwrk,
+ reps,ierr)
c% calling sequence
c subroutine n2solv(a,b,c,d,nwrk1,ndim,n,mdim,m,ind,ipr,niwrk,
c + reps,ierr)
c integer i,i1,ierr,ind,nwrk1,niwrk,irow2,j,j1,j2,k,lrow2,m,m1
c integer mdim,n,ndim,mfin,ipr(niwrk)
c double precision a(mdim,m),b(ndim,n),c(mdim,n),d(nwrk1),reps
c%purpose
c this routine is only to be called from syhsc
c%
c
integer i,i1,ierr,ind,nwrk1,niwrk,irow2,j,j1,j2,k,lrow2,m,m1
integer mdim,n,ndim,mfin,ipr(niwrk)
double precision a(mdim,m),b(ndim,n),c(mdim,n),d(nwrk1),reps
c
if(ind.lt.n-1) call backs2(c,b,ind,n,m,mdim,ndim)
c
m2=2*m
mfin=(m2*(m2+1))/2+4*m
do 20 i=1,m
m1=irow2(2*i-1,m)
k=lrow2(2*i-1,m)
i1=i-1
if(i.eq.1) i1=1
do 10 j=i1,m
j1=2*(j-i1+1)+m1
j2=1
if(m1.eq.0) j2=0
j2=j1+k-j2
d(j1-1)=a(i,j)
d(j1)=0.0d+0
d(j2)=a(i,j)
d(j2-1)=0.0d+0
10 continue
j1=m1+3
if(i.eq.1) j1=j1-2
d(j1)=d(j1)+b(ind,ind)
d(j1+1)=d(j1+1)+b(ind,ind+1)
if(i.ne.1) j1=m1+2
j1 = j1 + k
d(j1)=d(j1)+b(ind+1,ind)
d(j1+1)=d(j1+1)+b(ind+1,ind+1)
ip = 2*i + mfin
d(ip)=c(i,ind+1)
d(ip-1)=c(i,ind)
20 continue
c
call h2solv(d,nwrk1,ipr,niwrk,m,reps,ierr)
c
if(ierr.ne.0) go to 40
do 30 i=1,m
c(i,ind)=d(ipr(2*i-1))
c(i,ind+1)=d(ipr(2*i))
30 continue
ind=ind-2
return
40 ierr=-ind-1
return
end
subroutine h2solv(d,nwrk1,ipr,niwrk,m,reps,ierr)
integer i,i1,i2,ierr,ipr(niwrk),irow2,j,j1,k,k1,l,m21,mfin
double precision d(nwrk1),ddmax,reps
ierr=0
m2=2*m
mfin=(m2*(m2+1))/2+4*m
do 10 i=1,m2
ipr(m2+i)=irow2(i,m)
ipr(i)=i+mfin
10 continue
m21=m2-1
do 40 i=1,m21
i1=2
if(i.eq.m21) i1=1
l=0
ddmax=abs(d(ipr(m2+i)+1))
do 20 j=1,i1
if(abs(d(ipr(m2+j+i)+1)).le.ddmax)go to 20
ddmax=abs(d(ipr(m2+j+i)+1))
l=j
20 continue
if(ddmax.le.reps)go to 80
if(l.eq.0) go to 30
k=ipr(m2+i)
ipr(m2+i)=ipr(m2+l+i)
ipr(m2+l+i)=k
k=ipr(i)
ipr(i)=ipr(i+l)
ipr(i+l)=k
30 continue
ipr(m2+i+1)=ipr(m2+i+1)+1
if(i.ne.m21) ipr(m2+i+2)=ipr(m2+i+2)+1
ip1=i+1
do 40 j=1,i1
ddmax=d(ipr(m2+i+j))/d(ipr(m2+i)+1)
d(ipr(i+j))=d(ipr(i+j))-ddmax*d(ipr(i))
do 40 k1=ip1,m2
k=k1-i
d(ipr(m2+i+j)+k)=d(ipr(m2+i+j)+k)-
* ddmax*d(ipr(m2+i)+1+k)
40 continue
if(abs(d(ipr(m2+m2)+1)).le.reps) go to 80
d(ipr(m2))=d(ipr(m2))/d(ipr(m2+m2)+1)
do 60 i1=1,m21
i=m2-i1
i2=i+1
ddmax=0.0d+0
do 50 j1=i2,m2
j=j1-i2+2
ddmax=ddmax+d(ipr(j1))*d(ipr(m2+i)+j)
50 continue
d(ipr(i))=(d(ipr(i))-ddmax)/d(ipr(m2+i)+1)
60 continue
70 return
80 ierr=-1
go to 70
end
subroutine backs2(c,b,ind,n,m,mdim,ndim)
integer i,ind,ind1,ind2,j,m,mdim,n,ndim
double precision b(ndim,n),c(mdim,n)
ind1=ind+1
ind2=ind+2
do 10 i=ind2,n
do 10 j=1,m
c(j,ind1)=c(j,ind1)-b(ind1,i)*c(j,i)
c(j,ind)=c(j,ind)-b(ind,i)*c(j,i)
10 continue
return
end
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