By J. J. Dongarra, J. R. Bunch, G. B. Moler, G. W. Stewart
The authors of this conscientiously established consultant are the vital builders of LINPACK, a special package deal of Fortran subroutines for examining and fixing a variety of structures of simultaneous linear algebraic equations and linear least squares difficulties.
This advisor helps either the informal person of LINPACK who easily calls for a library subroutine, and the expert who needs to switch or expand the code to address particular difficulties. it's also instructed for school room paintings.
Adopted for lecture room use via Stanford college, Drexel collage, Rice college, and Pennsylvania kingdom collage (partial listing).
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Extra resources for LINPACK Users' Guide
Roundoff errors in floating point arithmetic operations usually cause the quantities computed by the subroutines in this chapter to be somewhat inaccurate. The following fairly vague statements give a general idea of the extent of these inaccuracies. SGEFA produces matrices L and U for which the product LU is almost always within roundoff error of A , no matter how close A is to being singular. SGESL produces a vector x for which Ax is almost always within roundoff error of b , no matter how close A is to being singular.
This subroutine is not included in LINPACK itself for several reasons. The corresponding double precision and complex subroutines would require quadruple precision and complex double precision dot products, respectively. Such extended precision cannot be obtained readily on many computers and cannot be efficiently simulated in standard Fortran. The extra storage requirements make the technique fairly costly for large matrices. The availability of RCOND partially obviates the need for the estimate provided by RELERR .
Let b be the vector of floating numbers actually input to SGESL and let x be the resulting computed solution, also a vector of floating point numbers. It is possible to prove that where a is another growth coefficient comparable to p . n is not too large, this inequality says that the size of the residual, b - Ax , is on the order of roundoff errors in A and x , even if <(A) is large. If, in addition, A is nonsingular, let x^ = A b be the exact answer. Then it is possible to prove that This says that if <(A) is large, then the relative error in x may also be quite large.
LINPACK Users' Guide by J. J. Dongarra, J. R. Bunch, G. B. Moler, G. W. Stewart