subroutine cpbco(abd,lda,n,m,rcond,z,info)
integer lda,n,m,info
complex abd(lda,1),z(1)
real rcond
c
c cpbco factors a complex hermitian positive definite matrix
c stored in band form and estimates the condition of the matrix.
c
c if rcond is not needed, cpbfa is slightly faster.
c to solve a*x = b , follow cpbco by cpbsl.
c to compute inverse(a)*c , follow cpbco by cpbsl.
c to compute determinant(a) , follow cpbco by cpbdi.
c
c on entry
c
c abd complex(lda, n)
c the matrix to be factored. the columns of the upper
c triangle are stored in the columns of abd and the
c diagonals of the upper triangle are stored in the
c rows of abd . see the comments below for details.
c
c lda integer
c the leading dimension of the array abd .
c lda must be .ge. m + 1 .
c
c n integer
c the order of the matrix a .
c
c m integer
c the number of diagonals above the main diagonal.
c 0 .le. m .lt. n .
c
c on return
c
c abd an upper triangular matrix r , stored in band
c form, so that a = ctrans(r)*r .
c if info .ne. 0 , the factorization is not complete.
c
c rcond real
c an estimate of the reciprocal condition of a .
c for the system a*x = b , relative perturbations
c in a and b of size epsilon may cause
c relative perturbations in x of size epsilon/rcond .
c if rcond is so small that the logical expression
c 1.0 + rcond .eq. 1.0
c is true, then a may be singular to working
c precision. in particular, rcond is zero if
c exact singularity is detected or the estimate
c underflows. if info .ne. 0 , rcond is unchanged.
c
c z complex(n)
c a work vector whose contents are usually unimportant.
c if a is singular to working precision, then z is
c an approximate null vector in the sense that
c norm(a*z) = rcond*norm(a)*norm(z) .
c if info .ne. 0 , z is unchanged.
c
c info integer
c = 0 for normal return.
c = k signals an error condition. the leading minor
c of order k is not positive definite.
c
c band storage
c
c if a is a hermitian positive definite band matrix,
c the following program segment will set up the input.
c
c m = (band width above diagonal)
c do 20 j = 1, n
c i1 = max0(1, j-m)
c do 10 i = i1, j
c k = i-j+m+1
c abd(k,j) = a(i,j)
c 10 continue
c 20 continue
c
c this uses m + 1 rows of a , except for the m by m
c upper left triangle, which is ignored.
c
c example.. if the original matrix is
c
c 11 12 13 0 0 0
c 12 22 23 24 0 0
c 13 23 33 34 35 0
c 0 24 34 44 45 46
c 0 0 35 45 55 56
c 0 0 0 46 56 66
c
c then n = 6 , m = 2 and abd should contain
c
c * * 13 24 35 46
c * 12 23 34 45 56
c 11 22 33 44 55 66
c
c linpack. this version dated 08/14/78 .
c cleve moler, university of new mexico, argonne national lab.
c
c subroutines and functions
c
c linpack cpbfa
c blas caxpy,cdotc,csscal,scasum
c fortran abs,aimag,amax1,cmplx,conjg,max0,min0,real
c
c internal variables
c
complex cdotc,ek,t,wk,wkm
real anorm,s,scasum,sm,ynorm
integer i,j,j2,k,kb,kp1,l,la,lb,lm,mu
c
complex zdum,zdum2,csign1
real cabs1
cabs1(zdum) = abs(real(zdum)) + abs(aimag(zdum))
csign1(zdum,zdum2) = cabs1(zdum)*(zdum2/cabs1(zdum2))
c
c find norm of a
c
do 30 j = 1, n
l = min0(j,m+1)
mu = max0(m+2-j,1)
z(j) = cmplx(scasum(l,abd(mu,j),1),0.0e0)
k = j - l
if (m .lt. mu) go to 20
do 10 i = mu, m
k = k + 1
z(k) = cmplx(real(z(k))+cabs1(abd(i,j)),0.0e0)
10 continue
20 continue
30 continue
anorm = 0.0e0
do 40 j = 1, n
anorm = amax1(anorm,real(z(j)))
40 continue
c
c factor
c
call cpbfa(abd,lda,n,m,info)
if (info .ne. 0) go to 180
c
c rcond = 1/(norm(a)*(estimate of norm(inverse(a)))) .
c estimate = norm(z)/norm(y) where a*z = y and a*y = e .
c the components of e are chosen to cause maximum local
c growth in the elements of w where ctrans(r)*w = e .
c the vectors are frequently rescaled to avoid overflow.
c
c solve ctrans(r)*w = e
c
ek = (1.0e0,0.0e0)
do 50 j = 1, n
z(j) = (0.0e0,0.0e0)
50 continue
do 110 k = 1, n
if (cabs1(z(k)) .ne. 0.0e0) ek = csign1(ek,-z(k))
if (cabs1(ek-z(k)) .le. real(abd(m+1,k))) go to 60
s = real(abd(m+1,k))/cabs1(ek-z(k))
call csscal(n,s,z,1)
ek = cmplx(s,0.0e0)*ek
60 continue
wk = ek - z(k)
wkm = -ek - z(k)
s = cabs1(wk)
sm = cabs1(wkm)
wk = wk/abd(m+1,k)
wkm = wkm/abd(m+1,k)
kp1 = k + 1
j2 = min0(k+m,n)
i = m + 1
if (kp1 .gt. j2) go to 100
do 70 j = kp1, j2
i = i - 1
sm = sm + cabs1(z(j)+wkm*conjg(abd(i,j)))
z(j) = z(j) + wk*conjg(abd(i,j))
s = s + cabs1(z(j))
70 continue
if (s .ge. sm) go to 90
t = wkm - wk
wk = wkm
i = m + 1
do 80 j = kp1, j2
i = i - 1
z(j) = z(j) + t*conjg(abd(i,j))
80 continue
90 continue
100 continue
z(k) = wk
110 continue
s = 1.0e0/scasum(n,z,1)
call csscal(n,s,z,1)
c
c solve r*y = w
c
do 130 kb = 1, n
k = n + 1 - kb
if (cabs1(z(k)) .le. real(abd(m+1,k))) go to 120
s = real(abd(m+1,k))/cabs1(z(k))
call csscal(n,s,z,1)
120 continue
z(k) = z(k)/abd(m+1,k)
lm = min0(k-1,m)
la = m + 1 - lm
lb = k - lm
t = -z(k)
call caxpy(lm,t,abd(la,k),1,z(lb),1)
130 continue
s = 1.0e0/scasum(n,z,1)
call csscal(n,s,z,1)
c
ynorm = 1.0e0
c
c solve ctrans(r)*v = y
c
do 150 k = 1, n
lm = min0(k-1,m)
la = m + 1 - lm
lb = k - lm
z(k) = z(k) - cdotc(lm,abd(la,k),1,z(lb),1)
if (cabs1(z(k)) .le. real(abd(m+1,k))) go to 140
s = real(abd(m+1,k))/cabs1(z(k))
call csscal(n,s,z,1)
ynorm = s*ynorm
140 continue
z(k) = z(k)/abd(m+1,k)
150 continue
s = 1.0e0/scasum(n,z,1)
call csscal(n,s,z,1)
ynorm = s*ynorm
c
c solve r*z = w
c
do 170 kb = 1, n
k = n + 1 - kb
if (cabs1(z(k)) .le. real(abd(m+1,k))) go to 160
s = real(abd(m+1,k))/cabs1(z(k))
call csscal(n,s,z,1)
ynorm = s*ynorm
160 continue
z(k) = z(k)/abd(m+1,k)
lm = min0(k-1,m)
la = m + 1 - lm
lb = k - lm
t = -z(k)
call caxpy(lm,t,abd(la,k),1,z(lb),1)
170 continue
c make znorm = 1.0
s = 1.0e0/scasum(n,z,1)
call csscal(n,s,z,1)
ynorm = s*ynorm
c
if (anorm .ne. 0.0e0) rcond = ynorm/anorm
if (anorm .eq. 0.0e0) rcond = 0.0e0
180 continue
return
end