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****** |
i n p u t ******
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nrow.....is the order of the matrix | | |
c .
nbands.....indicates its bandwidth, i.e.,
c(i,j) = 0 for i-j .ge. nbands .
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w.....workarray of size (nbands,nrow) | | |
containing the nbands diago-
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nals in its rows, with the main diagonal in row | | |
1 . precisely,
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w(i,j) |
contains c(i+j-1,j), i=1,...,nbands, j=1,...,nrow.
for example, the interesting entries of a seven diagonal sym-
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metric matrix | | |
c of order 9 would be stored in w as
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all other entries of | | |
w not identified in this way with an en-
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try of |
c are never referenced .
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diag.....is a work array of length | | |
nrow .
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****** |
o u t p u t ******
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w.....contains the cholesky factorization | | |
c = l*d*l-transp, with
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w(1,i) containing | | |
1/d(i,i)
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and |
w(i,j) containing l(i-1+j,j), i=2,...,nbands.
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****** |
m e t h o d ******
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gauss elimination, adapted to the symmetry and bandedness of | | |
c , is
used .
near zero pivots are handled in a special way. the diagonal ele-
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ment c(n,n) = w(1,n) is saved initially in |
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diag(n), all n. at the n-
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th elimination step, the current pivot element, viz. | | |
w(1,n), is com-
pared with its original value, diag(n). if, as the result of prior
elimination steps, this element has been reduced by about a word
length, (i.e., if w(1,n)+diag(n) .le. diag(n)), then the pivot is de-
clared to be zero, and the entire n-th row is declared to be linearly
dependent on the preceding rows. this has the effect of producing
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x(n) = 0 |
when solving c*x = b for x, regardless of b. justific-
ation for this is as follows. in contemplated applications of this
program, the given equations are the normal equations for some least-
squares approximation problem, diag(n) = c(n,n) gives the norm-square
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of the n-th basis function, and, at this point, | | |
w(1,n) contains the
norm-square of the error in the least-squares approximation to the n-
th basis function by linear combinations of the first n-1 . having
w(1,n)+diag(n) .le. diag(n) signifies that the n-th function is lin-
early dependent to machine accuracy on the first n-1 functions, there
fore can safely be left out from the basis of approximating functions
the solution of a linear system
c*x = b
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is effected by the succession of the following | | |
t w o calls:
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call bchfac ( w, nbands, nrow, diag ) | | |
, to get factorization
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call bchslv ( w, nbands, nrow, b ) | | |
, to solve for x.
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