河南大学数学与信息科学学院2011~2012学年第2学期期末考试.docVIP

河南大学数学与信息科学学院2011～20学年第学期期末考试 试卷 A卷 考试方式：闭卷 考试时间：120 分钟 卷面总分：100分 题 号 一 二 三 四 总成绩 合分人 得 分 and is approximated to , then has significant digits, and the absolute error accuracy is . 2 Suppose the formula is an interpolating type of numerical intergration, then ＝ . 3 If is Lagrange interpolating basis function, then we obtain that . 4 Given , then the spectral radiusof matrix A is , and the condition number is . 5 Suppose is an inner product space (内积空间), and is a finite dimension space, , then is an optimal approximation element to . 6 If and exists, and are two distinct interpolating points in ,then we have the remainder (余项) of cubic Hermite interpolation is . 得　分 评阅人 二、计算题，每题10分，共40分。 1 Using the 2th polynomial to fitting the following data by the Least Square method. x 1 2 3 4 y 4 10 18 26 2 Apply the Power method to solve the dominant eigenvalue (主特征值) of , that is, the eigenvalue with the largest magnitude, and the eigenvector associated with . 3 Assume that the numerical quadrature is Gaussian quadrature (高斯型积分), please determine A, B, C and (. 4 Give the following data: 0.30 0.40 0.55 0.65 0.80 1.05 0.30163 0.41075 0.57815 0.69675 0.87335 1.18885 Please use Lagrange interpolation and Piecewise linear Lagrange interpolation to estimate and . 得　分 评阅人 三、 证明题，每题12分，共24分。 1 Suppose the initial data is very close to （）, and the iterative formula is then the iterative sequence convergents to , and the order of convergence is 3. 得　分 评阅人 2 Given the linaer system of equations , where and . Show that Jacobi iterative method convergents, and Gauss-Seidel iterative method does not convergent. 四、综合题，每题12分，共12分。 Consider the first order initial-value problem (1) Show that the global truncation error of the Runge-Kutta method is (2)For the initial-value problem y(=-10y, , y(0)=1, taking the step