ELEMENTARY NUMERICAL ANALYSIS by ATKINSON HAN

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Elementary Numerical Analysis

GAUSSIAN NUMERICAL INTEGRATION 1. Show that if an integration formula of the form In ( f ) = ∑ wjf(xj) is exact when integrating 1, x, x2, …, xm, then it is exact for all polynomials of degree ≤ m. Please see attached for proper format of question.

Elementary Numerical Analysis

GAUSSIAN NUMERICAL INTEGRATION 1. Consider approximating integrals of the form... in which f(x) has several continuous derivatives on [0, 1] a. Find a formula... which is exact if f(x) is any linear polynomial. b. To find a formula... which is exact for all polynomial of degree ≤ 3, set up a system of four equations with unknown w1, w2, x1, x2. Verify that... Is a solution of the system. The verification can be numerical (say using Matlab) or symbolic (say using Maple or Mathematica). c. Apply I1 and I2 to the evaluation of... Please see attached.

Gaussian Quadrature

1. Consider approximating integrals of the form I ( f ) = ∫ √x f(x)dx in which f(x) has several continuous derivatives on [0, 1] a. Find a formula ∫ √x f(x)dx ≈ w1 f(x1) ≡ I1( f ) which is exact if f(x) is any linear polynomial. b. To find a formula ∫ √x f(x)dx ≈ w1 f(x1) + w2 f(x2) ≡ I2( f ) which is exact for all polynomial of degree ≤ 3, set up a system of four equations with unknown w1, w2, x1, x2 and find the values

Newton-Cotes formula

Derive all the weights for closed Newton-Cotes formula. Please see attached for full question.