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subroutine lybsc(n,a,na,c,x,u,eps,wrk,mode,ierr)
integer n,na,mode,ierr
double precision a(na,n),c(na,n),x(na,n),u(na,n),wrk(n)
double precision eps
C
C! calling sequence
C subroutine lybsc(n,a,na,c,x,u,wrk,mode,ierr)
C
C integer n,na,mode,ierr
C double precision a(na,n),c(na,n),x(na,n),u(na,n),wrk(n)
C
C arguments in
C
C
C n integer
C -the dimension of a
C
C a double precision(n,n)
C -coefficient matrix of the equation
C *** n.b. in this routine a is overwritten with
C its transformed upper triangular form (see comments)
C
C c double precision(n,n)
C -coefficient matrix of the equation
C
C na integer
C -the declared first dimension of a,c,x and u
C
C mode integer
C - mode = 0 if a has not already been reduced to
C upper triangular form
C - mode = 1 if a has been reduced to triangular form
C by (e.g) a previous call to lybsc
C
C arguments out
C
C a double precision(n,n)
C -on exit, a contains the transformed upper triangular
C form of a
C
C x double precision(n,n)
C -the solution matrix
C
C u double precision(n,n)
C - u contains the orthogonal matrix which was used
C to reduce a to upper triangular form
C
C ierr integer
C -error indicator
C
C ierr = 0 successful return
C
C ierr = 1 a has a degenerate pair of eigenvalues
C
C ierr = 2 a cannot be reduced to triangular form
C
C working space
C
C wrk double precision(n)
C
C!purpose
C
C to solve the double precision matrix equation
C
C trans(a)*x + x*a = c
C
C where c is symmetric, and trans(a) denotes
C the transpose of a .
C
C!method
C
C this routine is a modification of the subroutine atxxac,
C written and discussed by r.h.bartels & g.w.stewart.
C
C!reference
C
C r.h. bartels & g.w. stewart
C "solution of the matrix equation a'x + xb = c ",
C commun. a.c.m., vol 15, 1972, pp. 820-826 .
C
C!auxiliary routines
C
C orthes,ortran (eispack)
C sgefa,sgesl (linpack)
C lycsr (slice)
C
C!origin: adapted from
C
C control systems research group, dept eecs, kingston
C polytechnic, penrhyn rd.,kingston-upon-thames, england.
C
C!
C******************************************************************
C local variables:
C
integer i,j,k
double precision dprec,tt(1)
C
if (mode .eq. 1) goto 10
call orthes(na,n,1,n,a,wrk)
call ortran(na,n,1,n,a,wrk,u)
call hqror2(na,n,1,n,a,tt,tt,u,ierr,11)
if (ierr .ne. 0) goto 140
10 continue
do 20 i = 1,n
do 15 j = 1,n
x(i,j) = c(i,j)
15 continue
x(i,i) = x(i,i) * 0.50d+0
20 continue
C
do 40 i = 1,n
do 30 j = 1,n
dprec = 0.0d+0
do 25 k = i,n
dprec = dprec + x(i,k)*u(k,j)
25 continue
wrk(j) = dprec
30 continue
do 40 j = 1,n
x(i,j) = wrk(j)
40 continue
do 60 j = 1,n
do 50 i = 1,n
dprec = 0.0d+0
do 45 k = 1,n
dprec = dprec + u(k,i)*x(k,j)
45 continue
wrk(i) = dprec
50 continue
do 60 i = 1,n
x(i,j) = wrk(i)
60 continue
C
do 70 i = 1,n
do 70 j = i,n
x(i,j) = x(i,j) + x(j,i)
x(j,i) = x(i,j)
70 continue
C
C call shrslv (c,a,x,n,n,na,na,na,0.0d+0,1.0d+20,fail)
call lycsr(n,a,na,x,ierr)
if (ierr .ne. 0) return
C
do 80 i = 1,n
x(i,i) = x(i,i) * 0.50d+0
80 continue
C
do 100 i = 1,n
do 90 j = 1,n
dprec = 0.0d+0
do 85 k = i,n
dprec = dprec + x(i,k)*u(j,k)
85 continue
wrk(j) = dprec
90 continue
do 100 j = 1,n
x(i,j) = wrk(j)
100 continue
C
do 120 j = 1,n
do 110 i = 1,n
dprec = 0.0d+0
do 105 k = 1,n
dprec = dprec + u(i,k)*x(k,j)
105 continue
wrk(i) = dprec
110 continue
do 120 i = 1,n
x(i,j) = wrk(i)
120 continue
C
do 130 i = 1,n
do 130 j = i,n
x(i,j) = x(i,j) + x(j,i)
x(j,i) = x(i,j)
130 continue
C
goto 150
140 ierr = 2
150 return
end
subroutine lycsr(n,a,na,c,ierr)
c%calling sequence
c subroutine lycsr(n,a,na,c,ierr)
c integer n,na,ierr
c double precision a(na,n),c(na,n)
c%purpose
c
c
c this routine solves the continuous lyapunov equation where
c the matrix a has been transformed to quasi-triangular form.
c
c this routine is intended to be called only from
c slice routine lybsc .
c%
integer i,j,k,l,n,dk,dl,ia,kk,ll,na,job,km1,ldl,ierr,info,
x ldim,nsys,ipvt(4)
double precision a(na,n),c(na,n)
double precision t(4,4),p(4)
double precision dprec
ierr = 0
ldim = 4
job = 0
l = 1
10 dl = 1
if (l .eq. n) go to 20
if (a(l+1,l) .ne. 0.0d+0) dl = 2
20 ll = l + dl - 1
k = l
30 km1 = k - 1
dk = 1
if (k .eq. n) go to 34
if (a(k+1,k) .ne. 0.0d+0) dk = 2
34 kk = k + dk - 1
if (k .eq. l) go to 45
c
do 40 i = k, kk
c
do 40 j = l, ll
dprec = 0.0d+0
do 35 ia = l, km1
dprec = dprec + a(ia,i) * c(ia,j)
35 continue
c(i,j) = c(i,j) - dprec
40 continue
c
45 if (dl .eq. 2) go to 60
if (dk .eq. 2) go to 50
t(1,1) = a(k,k) + a(l,l)
if (t(1,1) .eq. 0.0d+0) go to 130
c(k,l) = c(k,l) / t(1,1)
go to 90
50 t(1,1) = a(k,k) + a(l,l)
t(1,2) = a(kk,k)
t(2,1) = a(k,kk)
t(2,2) = a(kk,kk) + a(l,l)
p(1) = c(k,l)
p(2) = c(kk,l)
nsys = 2
call dgefa(t,ldim,nsys,ipvt,info)
if (info .ne. 0) go to 130
call dgesl(t,ldim,nsys,ipvt,p,job)
c(k,l) = p(1)
c(kk,l) = p(2)
go to 90
60 if (dk .eq. 2) go to 70
t(1,1) = a(k,k) + a(l,l)
t(1,2) = a(ll,l)
t(2,1) = a(l,ll)
t(2,2) = a(k,k) + a(ll,ll)
p(1) = c(k,l)
p(2) = c(k,ll)
nsys = 2
call dgefa(t,ldim,nsys,ipvt,info)
if (info .ne. 0) go to 130
call dgesl(t,ldim,nsys,ipvt,p,job)
c(k,l) = p(1)
c(k,ll) = p(2)
go to 90
70 if (k .ne. l) go to 80
t(1,1) = a(l,l)
t(1,2) = a(ll,l)
t(1,3) = 0.0d+0
t(2,1) = a(l,ll)
t(2,2) = a(l,l) + a(ll,ll)
t(2,3) = t(1,2)
t(3,1) = 0.0d+0
t(3,2) = t(2,1)
t(3,3) = a(ll,ll)
p(1) = c(l,l) * 0.50d+0
p(2) = c(ll,l)
p(3) = c(ll,ll) * 0.50d+0
nsys = 3
call dgefa(t,ldim,nsys,ipvt,info)
if (info .ne. 0) go to 130
call dgesl(t,ldim,nsys,ipvt,p,job)
c(l,l) = p(1)
c(ll,l) = p(2)
c(l,ll) = p(2)
c(ll,ll) = p(3)
go to 90
80 t(1,1) = a(k,k) + a(l,l)
t(1,2) = a(kk,k)
t(1,3) = a(ll,l)
t(1,4) = 0.0d+0
t(2,1) = a(k,kk)
t(2,2) = a(kk,kk) + a(l,l)
t(2,3) = 0.0d+0
t(2,4) = t(1,3)
t(3,1) = a(l,ll)
t(3,2) = 0.0d+0
t(3,3) = a(k,k) + a(ll,ll)
t(3,4) = t(1,2)
t(4,1) = 0.0d+0
t(4,2) = t(3,1)
t(4,3) = t(2,1)
t(4,4) = a(kk,kk) + a(ll,ll)
p(1) = c(k,l)
p(2) = c(kk,l)
p(3) = c(k,ll)
p(4) = c(kk,ll)
nsys = 4
call dgefa(t,ldim,nsys,ipvt,info)
if (info .ne. 0) go to 130
call dgesl(t,ldim,nsys,ipvt,p,job)
c(k,l) = p(1)
c(kk,l) = p(2)
c(k,ll) = p(3)
c(kk,ll) = p(4)
90 k = k + dk
if (k .le. n) go to 30
ldl = l + dl
if (ldl .gt. n) return
c
do 120 j = ldl, n
c
do 100 i = l, ll
c(i,j) = c(j,i)
100 continue
c
do 120 i = j, n
dprec = 0.0d+0
do 110 k = l, ll
dprec = dprec + c(i,k) * a(k,j)
dprec = dprec + a(k,i) * c(k,j)
110 continue
c(i,j) = c(i,j) - dprec
c(j,i) = c(i,j)
120 continue
c
l = ldl
go to 10
c
130 ierr = 1
return
end
|
