西安建筑科技大学：《高等数学计算方法》课程教学资源（教材讲义）PaperA and Model Answe_PaperA

Anhui University Semester 1, 2004-2005 Final Examination(Paper A Subject title Time allowed: 2 hours School or Department: Ma Student Name: Student number Seat Number Question No.12345678910 11 Referee Score Marks In question 1-6, please choose the correct answer(only one is correct) 1.(5 marks)Word” MATLAB” comes form (A Mathematics Laborator (B)Matrix Laboratory (C) Mathematica Laboratory (D) Maple Laboratory 2.(5 marks) The matrix 215 (A) a strictly diagonally dominant matrix; (B)not a strictly diagonally dominant matrix (C)a singular matrix; (D) a matrix whose determinant is equal to zero 3.(5 marks) The computational complexity of Gaussian elimination for solv- ing linear equation systems AX=B(A is N X N matrix)is (A)O(N),(B)O(N2),(C)O(N),(D)O(N) 4.(5 marks) Assume that f(r)is defined on [a, b, which contains equally spaced nodes Tk=.o+hk that f"(a) a,b polation polynomial P(x)=∑f(xk)L1.(x) to approximate f(r), then the error E1(r)is (A)O(1)(B)O(h)(C)O(h2)(D)O(h3) 5.(5 marks) Degree 4 Chebyshev Polynomial T4( )is

Anhui University Semester 1, 2004-2005 Final Examination (Paper A) Subject title: Time allowed: 2 hours School or Department: Major: Student Name: Student Number: Seat Number: Question No. 1 2 3 4 5 6 7 8 9 10 11 Referee Score Marks In question 1-6, please choose the correct answer (only one is correct) 1. (5 marks) Word ”MATLAB” comes form (A) Mathematics Laboratory (B) Matrix Laboratory (C) Mathematica Laboratory (D) Maple Laboratory 2. (5 marks) The matrix   4 −1 1 4 −8 1 −2 1 5   is (A) a strictly diagonally dominant matrix; (B) not a strictly diagonally dominant matrix; (C) a singular matrix; (D) a matrix whose determinant is equal to zero. 3. (5 marks) The computational complexity of Gaussian elimination for solv￾ing linear equation systems AX = B (A is N × N matrix) is (A) O(N), (B) O(N 2 ), (C) O(N 3 ), (D) O(N 4 ). 4. (5 marks) Assume that f(x) is defined on [a, b], which contains equally spaced nodes xk = x0 + hk. Additionally, assume that f 00(x) is continuous on [a, b]. If we use Lagrange interpolation polynomial P1(x) = X 1 k=0 f(xk)L1,k(x) to approximate f(x), then the error E1(x) is (A) O(1) (B) O(h) (C) O(h 2 ) (D) O(h 3 ) 5. (5 marks) Degree 4 Chebyshev Polynomial T4(x) is 1

a an odd function (B)an even function (C) not odd or even function,(D) both odd and even functi 6.(5 marks)Pade Approximation is (A) A rational polynomial approximation (B)A polynomial approximation (C) A triangular polynomial approximation (D) A linear function approximation 7.(15 marks) Use the false position method to find the root of x sin(a)-1=0 that is located in the interval 0, 2(the function sin(a)is evaluated in radians)

(A) an odd function, (B) an even function (C) not odd or even function, (D) both odd and even function 6. (5 marks) Pad´e Approximation is (A) A rational polynomial approximation (B) A polynomial approximation (C) A triangular polynomial approximation (D) A linear function approximation 7. (15 marks) Use the false position method to find the root of x sin(x)−1 = 0 that is located in the interval [0, 2] (the function sin(x) is evaluated in radians). 2

8.(15 marks) In the following linear equation systems 4x-y+z=7 4x-8y+z=-21 2x+y-52=15, tart with Po=(1, 2, 2), and use Gauss-Seidel iteration to find Pk for k= 1, 2 Will Gauss-Seidel iteration converge to the solution? 9.(15 marks) Consider y= f(r)= cos(a)over [0.0, 1.2. Use the three nodes o=0.0, 1=0.6, and 2= 1.2 to construct a quadratic interpolation polynomial P2(r)

8. (15 marks) In the following linear equation systems 4x − y + z = 7 4x − 8y + z = −21 −2x + y − 5z = 15, start with P0 = (1, 2, 2), and use Gauss-Seidel iteration to find Pk for k = 1, 2. Will Gauss-Seidel iteration converge to the solution? 9. (15 marks) Consider y = f(x) = cos(x) over [0.0, 1.2]. Use the three nodes x0 = 0.0, x1 = 0.6, and x2 = 1.2 to construct a quadratic interpolation polynomial P2(x). 3

10.(15 marks) Consider f(ar)=2+sin(2v). Use the composite Simpson rule with 1l sample points to compute an approximation to the integral of f(a)taken over [1, 6 11.(10 marks)Assume that g E Cla, b. If the range of the mapping y=g(a)satisfies y E a, b for all x E [a, b, then g has a fixed point in [a, b

10. (15 marks) Consider f(x) = 2+ sin(2√ x). Use the composite Simpson rule with 11 sample points to compute an approximation to the integral of f(x) taken over [1, 6]. 11. (10 marks) Assume that g ∈ C[a, b]. If the range of the mapping y = g(x) satisfies y ∈ [a, b] for all x ∈ [a, b], then g has a fixed point in [a, b]. 4