tailieunhanh - numerical mathematics and scientific computation volume 1 Episode 2

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 30 Chapter 1. Principles of Numerical Calculations 2. Show how repeated synthetic division can be used to move the origin of a polynomial . given 1 2 an and z find Cl C2 . Cn so that Cj x zý ỵ. Write a program for synthetic division with this ordering of the coefficients and apply it to this algorithm. Hint Set Pn x Apply synthetic division to Pn x p -i x pn x -pn z x - z etc. 3. a Show that the transformation made in Problem 2 can also be expressed by means of the matrix-vector equation c diag 1-i Fdiag 7-1 a where a 1 2 a i T c ci C2 . C T and diag 7-1 is a diagonal matrix with the elements c7-1 j 1 n. The matrix p is of size n n its elements are Pij if j i else Pij 0. By convention 0 1 here. b Note the relation of p to the Pascal triangle and show how p can be generated by a simple recursion formula. Also show how each element of F-1 can be expressed in terms of the corresponding element of F. How is the origin of the polynomial Pn x moved if you replace F by F-1 in the matrix-vector equation that defines c c If you reverse the order of the elements of the vectors a c this may sometimes be a more convenient ordering how is the matrix F changed Comment With a terminology to be used much in this book we can look upon a and c as different coordinate vectors for the same element in the n-dimensional linear space Vn of polynomials of degree less than n. The matrix F gives the coordinate transformation. 4. Derive recurrence relations and write a program for computing the coefficients of the product r of two polynomials p and q r x p x l x m n m n 1 f 2aixi 1j 2bjXj 1 j c -1- i l z J 1 k 1 5. Let X y be nonnegative integers with y 0 0. The division x y yields the quotient q and the remainder r. Show that if X and y have a common factor then that number is a divisor of r as well. Use this remark to design an algorithm for the determination of the greatest common divisor of X and y Euclid s algorithm . 6. Derive a forward and a backward recurrence relation for .