Research On Accurate Algorithm Based On Multiple-component Format Of Floating-point Numbers

As the supercomputers develop rapidly,high-performance computing technology faces great challenges.Since most computers use floating-point arithmetic,it is important to reduce the round-off error in the numerical calculation for large-scale problems.Based on the forward-backward error analysis theory,the running error analysis theory,error-free transformation technology and double-double arithmetic,this thesis proposes several compensated algorithms based on multiple-component format,including quotient-difference algorithm,Clenshaw-Smith algorithm,Barrio-Clenshaw-Smith algorithm,Horner algorithm,Volk-Schumaker algorithm,and their error analysis,numerical experiments and applications.In this thesis,we have also developed some high-precision library by multiple-component format in C language,Matlab and Scilab.The main work of this thesis can be divided into three parts:(1)The first part(chapter 2)makes a stability analysis of the quotient-difference algorithm and designs a new compensated quotient-difference algorithm.The quotientdifference algorithm can be applied to locate poles or zeros of polynomial,páde approximation and eigenvalue problems,but it is very unstable.We provide a condition number for quotient-difference algorithm,design the compensated quotient-difference algorithm by error-free transform and compensated division algorithm,and give the stability analysis of quotient-difference algorithm and its compensated algorithm through forwardbackward error analysis theory.The numerical experiments show the validity of the compensation quotient-difference algorithm and its forward error bound.Accuracy of the compensated algorithm is also tested by three applications.(2)The second part(chapter 3,4)focuses on the problem of evaluating polynomial series,including one-dimensional classical orthogonal polynomial series and twodimensional tensor product surface.Evaluation of classical orthogonal polynomial series and its k-th derivative need to use Clenshaw-Smith algorithm and Barrio-ClenshawSmith algorithm,Meanwhile,evaluation of two-dimensional polynomial series in power basis needs bivariate Horner algorithm,and evaluation of Bézier tensor product surface needs two-dimensional Volk-Schumaker algorithm.Applying error-free transformation technology,we design some compensated algorithms of these algorithms.Their stability analysis and running error bounds are also given by forward-backward error analysis theory and running error analysis theory.At the same time,the numerical experiment gives the validity verification of the compensated algorithms,forward error bounds and running error bounds.Moreover,all algorithms for evaluating classical orthogonal polynomial series and their k-th derivative,their compensated algorithms and their running error bounds are integrated into C and Matlab platforms which form a high-precision library.Numerical tests with some practical applications verifies that the algorithms in this library are accurate,effective and reasonable.(3)The third part(chapter 5)designs a package of mixed extension precision.In this part,double-double precision,triple-double precision and quadruple-double precision are defined as the extended precision.Combining with real and complex numbers,we provide the mixing extension precision package with a number of algorithms.Meanwhile,in Matlab or Scilab,we redefine the class of extension precision,and overload all kinds of arithmetic operation.We also design two display modes and two accuracy modes through initialization.Finally,numerical experiments test the accuracy and speed of this software package.