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subroutine qhesz(nm,n,a,b,matq,q,matz,z)
c
integer i,j,k,l,n,lb,l1,nm,nk1,nm1,nm2
double precision a(nm,n),b(nm,n),z(nm,n),q(nm,n)
double precision r,s,t,u1,u2,v1,v2,rho
logical matz,matq
c
c! purpose
c this subroutine accepts a pair of real general matrices and
c reduces one of them to upper hessenberg form and the other
c to upper triangular form using orthogonal transformations.
c it is usually followed by qzit, qzval and, possibly, qzvec.
c
c! calling sequence
c
c subroutine qhesz(nm,n,a,b,matq,q,matz,z)
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 matrices;
c
c a contains a real general matrix;
c
c b contains a real general matrix;
c
c matz should be set to .true. if the right hand transformations
c are to be accumulated for later use in computing
c eigenvectors, and to .false. otherwise.
c
c on output:
c
c a has been reduced to upper hessenberg form. the elements
c below the first subdiagonal have been set to zero;
c
c b has been reduced to upper triangular form. the elements
c below the main diagonal have been set to zero;
c
c z contains the product of the right hand transformations if
c matz has been set to .true. otherwise, z is not referenced.
c
c! originator
c
c this subroutine is the first step of the qz algorithm
c for solving generalized matrix eigenvalue problems,
c siam j. numer. anal. 10, 241-256(1973) by moler and stewart.
c (modification de la routine qzhes de eispack pour avoir
c la matrice unitaire de changement de base sur les lignes
c donne par la matrice q .memes conventions que pour z.)
c f.d.
c!
c questions and comments should be directed to b. s. garbow,
c applied mathematics division, argonne national laboratory
c
c ------------------------------------------------------------------
c
c :::::::::: initialize z ::::::::::
if (.not. matz) go to 10
c
do 3 i = 1, n
c
do 2 j = 1, n
z(i,j) = 0.0d+0
2 continue
c
z(i,i) = 1.0d+0
3 continue
10 continue
if(.not.matq) goto 11
do 31 i=1,n
do 21 j=1,n
q(i,j)=0.0d+0
21 continue
q(i,i)=1.0d+0
31 continue
11 continue
c :::::::::: reduce b to upper triangular form ::::::::::
if (n .le. 1) go to 170
nm1 = n - 1
c
do 100 l = 1, nm1
l1 = l + 1
s = 0.0d+0
c
do 20 i = l1, n
s = s + abs(b(i,l))
20 continue
c
if (s .eq. 0.0d+0) go to 100
s = s + abs(b(l,l))
r = 0.0d+0
c
do 25 i = l, n
b(i,l) = b(i,l) / s
r = r + b(i,l)**2
25 continue
c
r = sign(sqrt(r),b(l,l))
b(l,l) = b(l,l) + r
rho = r * b(l,l)
c
do 50 j = l1, n
t = 0.0d+0
c
do 30 i = l, n
t = t + b(i,l) * b(i,j)
30 continue
c
t = -t / rho
c
do 40 i = l, n
b(i,j) = b(i,j) + t * b(i,l)
40 continue
c
50 continue
c
do 80 j = 1, n
t = 0.0d+0
c
do 60 i = l, n
t = t + b(i,l) * a(i,j)
60 continue
c
t = -t / rho
c
do 70 i = l, n
a(i,j) = a(i,j) + t * b(i,l)
70 continue
c
80 continue
if(.not.matq) goto 99
do 780 j = 1, n
t = 0.0d+0
c
do 760 i = l, n
t = t + b(i,l) * q(i,j)
760 continue
c
t = -t / rho
c
do 770 i = l, n
q(i,j)=q(i,j)+t*b(i,l)
770 continue
c
780 continue
99 continue
c
b(l,l) = -s * r
c
do 90 i = l1, n
b(i,l) = 0.0d+0
90 continue
c
100 continue
c :::::::::: reduce a to upper hessenberg form, while
c keeping b triangular ::::::::::
if (n .eq. 2) go to 170
nm2 = n - 2
c
do 160 k = 1, nm2
nk1 = nm1 - k
c :::::::::: for l=n-1 step -1 until k+1 do -- ::::::::::
do 150 lb = 1, nk1
l = n - lb
l1 = l + 1
c :::::::::: zero a(l+1,k) ::::::::::
s = abs(a(l,k)) + abs(a(l1,k))
if (s .eq. 0.0d+0) go to 150
u1 = a(l,k) / s
u2 = a(l1,k) / s
r = sign(sqrt(u1*u1+u2*u2),u1)
v1 = -(u1 + r) / r
v2 = -u2 / r
u2 = v2 / v1
c
do 110 j = k, n
t = a(l,j) + u2 * a(l1,j)
a(l,j) = a(l,j) + t * v1
a(l1,j) = a(l1,j) + t * v2
110 continue
c
a(l1,k) = 0.0d+0
c
do 120 j = l, n
t = b(l,j) + u2 * b(l1,j)
b(l,j) = b(l,j) + t * v1
b(l1,j) = b(l1,j) + t * v2
120 continue
if(.not.matq) goto 122
do 121 j=1,n
t=q(l,j)+u2*q(l1,j)
q(l,j)=q(l,j)+t*v1
q(l1,j)=q(l1,j)+t*v2
121 continue
122 continue
c :::::::::: zero b(l+1,l) ::::::::::
s = abs(b(l1,l1)) + abs(b(l1,l))
if (s .eq. 0.0d+0) go to 150
u1 = b(l1,l1) / s
u2 = b(l1,l) / s
r = sign(sqrt(u1*u1+u2*u2),u1)
v1 = -(u1 + r) / r
v2 = -u2 / r
u2 = v2 / v1
c
do 130 i = 1, l1
t = b(i,l1) + u2 * b(i,l)
b(i,l1) = b(i,l1) + t * v1
b(i,l) = b(i,l) + t * v2
130 continue
c
b(l1,l) = 0.0d+0
c
do 140 i = 1, n
t = a(i,l1) + u2 * a(i,l)
a(i,l1) = a(i,l1) + t * v1
a(i,l) = a(i,l) + t * v2
140 continue
c
if (.not. matz) go to 150
c
do 145 i = 1, n
t = z(i,l1) + u2 * z(i,l)
z(i,l1) = z(i,l1) + t * v1
z(i,l) = z(i,l) + t * v2
145 continue
c
150 continue
c
160 continue
c
170 return
c :::::::::: last card of qzhes ::::::::::
end
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