数值分析试卷(英文版).docVIP

1.(20%) Fill in the following blanks. (1) Let =5.3001 be an approximation to =5.300186, then retains _______________ significant digits,and the relative error of is __________________. (2) Suppose , , then =__________, =___________. (3) Suppose is the root of the equation of multiplicity ,then the modified Newton’s iterative formula for finding is______________________________________________. (4) Let and the cubic Lagrange interpolating polynomial for on the nodes 0,1,2,3 is . Then the error =_______________________________, and =_____________________. (5) If , then the divided differences =____________, and =_____________. (6) Write an three-point formula to approximate by using of ,,:__________________________________________________. 2.(12%) Establish the Jacobi iterative scheme and Gauss-Seidel iterative scheme for the following linear system: Test the convergence of the Gauss_Seidel iterative scheme by using the norm of its iterative matrix. 3.(12%) (1) If the norm of the iterative matrix is , please poof the following formula: (2) Let and .Use Newton’s method to find . (3) Let ,and .Use the Secant method to find . 4.(12%) Use the following data to construct an interpolating polynomial of the degree three such that for ,and . 0 1 2 1 2 3 2 4 12 3 5.(12%) Suppose for . (a) Determine such that are orthogonal on with respect to the weight function . (b) Find the least squares approximating polynomial of degree two for the function on the interval 6.(12%) The Midpoint rule for the approximating gives the value 12,the Composite Midpoint rule with gives 5,and Composite Simpson’s rule gives 6.Use the fact that and to determine . 7.(10%) Determine constant that will produce a quadrature formula: that has degree of 3. 8.(10%) (1)Suppose exists on .Show that the following two-step explicit scheme is a scheme of order two to solve the initial value problem for the