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Find a formula for [ f(x)dx in terms of the functional values f(x₁) and f'(x2), which is exact for linear poly- nomials. For what values of x₁ or x2 is the discretization formula superaccurate? auplinit Crank Nicholson and implicit

Find a formula for [ f(x)dx in terms of the functional values f(x₁) and f'(x2), which is exact for linear poly- nomials. For what values of x₁ or x2 is the discretization formula superaccurate? auplinit Crank Nicholson and implicit

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section: Chapter Questions
Problem 5T
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S 0 f(x) dx. First derive a formula that is accurate for linear polynomials, then quadradic and finally cubic. Relate the resulting formulas to the Trapizodal and Simpson's Rules. 3. A discretization formula is called superaccurate when it uses fewer functional values f(xi) than expected. Show that if x₁ in the first case in Problem 2 then f(x2) does not have to be used. In this case the discretization formula is called the Midpoint Rule. Find the Values of x; for which the other two formulas in Problem 2 are superaccurate. 4. Find a formula for [ f(x) dx in terms of the functional values f(x₁) and f'(x2), which is exact for linear poly- nomials. For what values of x₁ or x2 is the discretization formula superaccurate? 5. Use the functional method to derive the explicit, Crank-Nicholson and implicit schemes for the heat equations with vanishing BC. Show that the errors are O(At, (Ax)²), 0((At)², (Ax)2) and O(At, (Ax)2), respectively (use the inter- polation theorem 10.1). You can do this by first holding x and then t fixed, explain why. C6. Write a scheme to find the nth degree Newtons polynomial that solves the inter- polation problem. Then write a scheme to implement the functional method to find the second derivative at a point. Let the accuracy be a choice. Make sure that your matrix is upper diagonal (this is why we use Newtons polynomial) so that you can solve the final system by backward substitution.
Transcribed Image Text:S 0 f(x) dx. First derive a formula that is accurate for linear polynomials, then quadradic and finally cubic. Relate the resulting formulas to the Trapizodal and Simpson's Rules. 3. A discretization formula is called superaccurate when it uses fewer functional values f(xi) than expected. Show that if x₁ in the first case in Problem 2 then f(x2) does not have to be used. In this case the discretization formula is called the Midpoint Rule. Find the Values of x; for which the other two formulas in Problem 2 are superaccurate. 4. Find a formula for [ f(x) dx in terms of the functional values f(x₁) and f'(x2), which is exact for linear poly- nomials. For what values of x₁ or x2 is the discretization formula superaccurate? 5. Use the functional method to derive the explicit, Crank-Nicholson and implicit schemes for the heat equations with vanishing BC. Show that the errors are O(At, (Ax)²), 0((At)², (Ax)2) and O(At, (Ax)2), respectively (use the inter- polation theorem 10.1). You can do this by first holding x and then t fixed, explain why. C6. Write a scheme to find the nth degree Newtons polynomial that solves the inter- polation problem. Then write a scheme to implement the functional method to find the second derivative at a point. Let the accuracy be a choice. Make sure that your matrix is upper diagonal (this is why we use Newtons polynomial) so that you can solve the final system by backward substitution.
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