Calculate sensitivity function using parallel algorithm.(Formula)(Technical report)

INTRODUCTION

The ability to develop mathematical models in Biology, Physics, Geology and other applied areas has pull and has been pushed by, the advances in High Performance Computing. Moreover, the use of iterative methods has increased substantially in many application areas in the last years (9). One reason for that is the advent of parallel Computing and its impact in the overall performance of various algorithms on numerical analysis (1). The use of clusters plays an important role in such scenario as one of the most effective manner to improve the computational power without increasing costs to prohibitive values. However, in some cases, the solution of numerical problems frequently presents accuracy issues increasing the need for computational power. Verified computing provides an interval result that surely contains the correct result. Numerical applications providing automatic result verification may be useful in many fields like simulation and modeling. Finding the verified result often increases dramatically the execution time (2). However, in some numerical problems, the accuracy is mandatory. The requirements for achieving this goal are: interval arithmetic, high accuracy combined with well suitable algorithms. The interval arithmetic defines the operations for interval numbers, such that the result is a new interval that contains the set of all possible solutions. The high accuracy arithmetic ensures that the operation is performed without rounding errors and rounded only once in the end of the computation. The requirements for this arithmetic are: the four basic operations with high accuracy, optimal scalar product and direct rounding. This arithmetic's should be used in appropriate algorithms to ensure that those properties will be hold. There is a multitude of tools that provide verified computing; among them an attractive option is C-XSC (C for extended Scientific Computing) (3). CXSC is a free and portable programming environment for C and C++ programming Languages, offering high accuracy and automatic verified results. This programming Tool allows the solution of several standard problems, including many reliable numerical parallel algorithms. The need to solve systems of linear algebraic equations arises frequently in scientific and engineering applications, with the solution being useful either by itself or as an intermediate step in solving a larger problem. In practical problems, the order, n, may in many cases be large (100-1000) or very large (many tens or hundreds of thousands). The cost of a numerical procedure is clearly an important consideration-so too is the accuracy of the method. Let us consider a system of linear algebraic equations:

AX = B (1)

Where:

A = [{aij}.sup.n] i,j = 1 is a given matrix

B = [(b1, ..., bn).sup.t] is a given vector

It is well known (for example (4), (5)) that the solution, x, x [member of] [R.sup.n], when it exists, can be found using-direct methods, such as Gaussian elimination and LU and Cholesky decomposition, …