(c) We want to find a three-point formula, [*^* ƒ (x)dx = w₁ f(x1) + w2f (x2) + wzf(xx) + E, -h similar to the two-point formula in part ((b)). X1 < x2 < X3, (3) Definite integrals of f(x) and f(x) on a symmetric interval should be equal, i.e. L f(x)dx = L f(x)dx. -h (4) The quadrature rule should also possess such a property of integrals. Using this ar- gument of symmetry, show that x2 = 0 and find relations between x₁ and x3; and between w₁ and w3. Then, calculate the weights and points that make this rule ex- act for polynomials of the highest degree possible. Find the error term and write the quadrature formula (3).
(c) We want to find a three-point formula, [*^* ƒ (x)dx = w₁ f(x1) + w2f (x2) + wzf(xx) + E, -h similar to the two-point formula in part ((b)). X1 < x2 < X3, (3) Definite integrals of f(x) and f(x) on a symmetric interval should be equal, i.e. L f(x)dx = L f(x)dx. -h (4) The quadrature rule should also possess such a property of integrals. Using this ar- gument of symmetry, show that x2 = 0 and find relations between x₁ and x3; and between w₁ and w3. Then, calculate the weights and points that make this rule ex- act for polynomials of the highest degree possible. Find the error term and write the quadrature formula (3).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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