数值分析试卷

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1、期末统考试卷A201- 201学年第学期 课程代码课程名称数值分析学分课程性质：必修囘选修口 限修口 考试形式：闭卷囘开卷口专业班级（教学班）考试日期 命题教师 系（所或教研室）主任审批签名3. (12%)(1) If the norm of the iterative matrix B is|B| q 1 , please poof the following1. (20%) Fill in the following blanks.(1) Let x* =5.3001 be an approximation to x =5.300186, then x* retains significa

2、ntx (k) x (k-1)digits,and the relative error of x* is-1 2 Suppose x (2,1,3,4)t, A _ 34_,then何1=，A2(3) Suppose x* is the root of the equation f (x) 0 of multiplicity m(m 2) ,then the modifiedqformula: Tx (k) x*| Let f (x)二 xex -1 and x0 二 0.5 g Newtons method to find x2 . Let f (x)二 xex 1, x0 二 0.5 a

3、nd x1 二 0.6 g the Secant method to find x3.Newtons iterative formula for finding x* is(4) Let f (x) x4 and the cubic Lagrange interpolating polynomial for f on the nodes 0,1,2,3 isP3( %).Then the error f (x) P (x)=3,and P3( %)= If f (x) _2x6 + 3x3 + 2x +1, then the divided differences f心，21,26=f 2o,

4、.,27 =(6) Write anO(h 2)three-point formula to approximatef (x0)by using off(x ),f(x + h),f (x + 2h)：0 0 04.(12%) Use the following data to construct an interpolating polynomialp3 (x) of the degree three such thatp3(xi)二 f(xJ for i = 0,1,2，and p3(x1)二 f (x1).i012xi123f (x)i2412f(x)i32. (12%) Establi

5、sh the Jacobi iterative scheme and Gauss-Seidel iterative scheme for the following linear system:4x - 2x + x 二 2,123J2x + 6x + 3x 二 3,123x + 2x + 4x 二 5.123Test the convergence of the Gauss_Seidel iterative scheme by using the norm of its iterative matrix.5.(12%) Suppose w( x)三 1, P (x)三 1, P (x) =

6、x B , P (x)二(x B ) P CP (x) for k 2.011 kkk1k k2(a) Determine B , B ,C such that P (x),P (x),P (x) are orthogonal on 1,1 with respect to the1 2 2 0 1 2weight function w(x).(b) Find the least squares approximating polynomial of degree two for the function f (x) x4 on the interval1,1期末统考试卷A201- 201学年第

7、学期专业班级（教学班）课程代码课程名称数值分析学分课程性质：必修囘选修口 限修口 考试形式：闭卷3 开卷口考试日期命题教师系（所或教研室）主任审批签名6(12%) The Midpoint rule for the approximating j 1 f (x)dx gives the value 12,the Composite Midpoint rule-1with n 2 gives 5,and Composite Simpsonrule gives 6.Use the fact that f (-1) f (1) andf (-0.5)二 f (0.5) -1 to determine

8、 f (-1), f (-5),f (0), f (0.5), f (1).7(10%) Determine constant a,b,c,d that will produce a quadrature formula:ji f (x)dx 二 af (-1) + bf (1) + cf (-1) + df (1) that has degree of 3.-18.(10%) (l)Suppose y(t) exists on (a，b) .Show that the following two-step explicit scheme广w a ,00w a ,- 11w w + h(3f (t, w) -f (t , w )i 1,2,3,n-1.i+1 i 2i ii-1i-1is a scheme of order two to solve the initial value problem for the ordinary differential equation: y (t ) a ,00彳 y(t ) a ,11y (t) f (t, y), t g a,bWhereh , t a + ih, i 0,1,2,八,nn i