The following data can be used to approximate the integral M = N 1 (h) = XXXXXXXXXX, N 1 (h/2) = XXXXXXXXXX, N 1 (h/4) = XXXXXXXXXX, N 1 (h/8) = XXXXXXXXXX Assume a formula exists of the type given M=...

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The following data can be used to approximate the integral
M = N1(h) = 2.356194, N1(h/2) = -0.4879837,
N1(h/4) = -0.8815732, N1(h/8) = -0.9709157
Assume a formula exists of the type given M= and determine N4(h).
8)
The forward-difference formula can be expressed as
f’(x0) = 1/h [ f(x0+h) – f(x0) ] – h/2 f’’(x0) – h2/6 f’’’(x0) + O(h3).
Use extrapolation to derive an O(h3) formula for f’ (x0).
10)
Suppose that N(h) is an approximation to M for every h>0 and that
M = N(h) + K1h2
+ K2h4
+K3h6
+ …,
for some constants K1, K2, K3, …. Use the values N(h), N(h/3) and N(h/9) to produce an O(h6) approximation to M.
Section 4.3
20)
Find the constants x0, x1, and c1
so that the quadrature formula
= ½ f(x0) + c1f (x1) has the highest possible degree of precision.
Section 4.5
10)
Romberg integration for approximating gives R11
= 4 and R22
= 5. Find f(1/2).



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Section 4.2 Numerical differentiation and integretation 6) The following data can be used to approximate the integral M = 03p/2cosxdx. N1(h) = 2.356194,N1(h/2) = -0.4879837, N1(h/4) = -0.8815732, N1(h/8) = -0.9709157 Assume a formula exists of the type given M= 0psinxdx and determine N4(h). 8) The forward-difference formula can be expressed as f’(x0) = 1/h [ f(x0+h) – f(x0) ] – h/2 f’’(x0) – h2/6 f’’’(x0) + O(h3). Use extrapolation to derive an O(h3) formula for f’ (x0). 10) Suppose that N(h) is an approximation to M for every h>0 and that M = N(h) + K1h2 + K2h4 +K3h6 + …, for some constants K1, K2, K3, …. Use the values N(h), N(h/3) and N(h/9) to produce an O(h6) approximation to M. Section 4.3 20) Find the constants x0, x1, and c1 so that the quadrature formula 01fxdx = ½ f(x0) + c1f (x1) has the highest possible degree of precision. Section 4.5 10) Romberg integration for approximating 01fxdx gives R11 = 4 and R22 = 5. Find f(1/2). 12) Use Romberg integration to compute the following approximations to 048[1+ (cosx)2] dx. Determine R1,1 , R2,1 , R3,1 , R4,1 and R5,1 and use these approximations to predict the value of the integral. Determine R2,2 , R3,3 , R4,4 and R5,5 and modify your prediction. Determine R6,1 , R6,2 , R6,3 , R6,4 , R6,5 and R6,6 and modify your prediction. Determine R7,7 , R8,8 , R9,9 and R10,10 and make a final prediction. Explain why this integral causes difficulty with Romberg integration and how it can be reformulated to more easily determine an accurate approximation. Section 4.6 Adaptive Quadrature Methods 4) Use Adaptive quadrature to approximate the following integrals to within 10-5. 13e2 sin3x dx 13e3x sin2x dxType equation here. 05(2x cosx2x- x-22) dx 05(4xcos2x- (x-2)2)dx 7) The differential equation mu’’(t) + ku(t) = F0 cos ?t Describes a spring-mass system with mass m, spring constant k, and no applied damping. The term F0 cos ?t describes a periodic external force applied to the...



Answered Same DayDec 20, 2021

Answer To: The following data can be used to approximate the integral M = N 1 (h) = XXXXXXXXXX, N 1 (h/2) =...

Robert answered on Dec 20 2021
129 Votes
8) f1(x0) = 1/h(f(x0+h)-f(x0)) – h/2f11(x0) – h2/6f111(x0) + O(h3)
8) f1(x0) = 1/h(f(x0+h)-f(x0)) – h/2f11(x0) – h2/6f111(x0) + O(h
3)……..(1)
In general, Richardson’s extrapolation is used to generate high-accuracy approximations while using low-order formulas.
Replacing h in (1) with 2h gives the new formula
f1(x0) = 1/2h(f(x0+2h)-f(x0)) – hf11(x0) – 4h2/6f111(x0) + O(h3)………(2)
Multiplying equation (1) by 2 and subtracting equation (2)
f1(x0) = 2/h(f(x0+h)-f(x0)) – 1/2h(f(x0+2h)-f(x0)) + h2/3f111(x0) + O(h3)
= 1/2h(-f(x0+2h)+4f(x0+h)-3f(x0)) + h2/3f111(x0) + O(h3)………(3)
Replacing h in (3) with 2h
f1(x0) = 1/4h(-f(x0+4h)+4f(x0+2h)-3f(x0)) + 4h2/3f111(x0) + O(h3)……(4)
Multiplying equation (3) by 4 and subtracting equation (4)
3f1(x0) =1/2h(-4f(x0+2h) + 16f(x0+h)-12f(x0)) -1/4h(-4f(x0+4h)+ 4f(x0+2h)-3f(x0)) + O(h3)
= 1/4h(f(x0+4h) - 12f(x0+2h)+32f(x0+h)-21f(x0)) + O(h3)
f1(x0) = 1/12h(f(x0+4h) - 12f(x0+2h)+32f(x0+h)-21f(x0)) + O(h3)
10) We would like to find coefficients A,B and C such that
AN(h) + BN(h/3) + CN(h/9) = M + O(h6)
AN(h) + BN(h/3) + CN(h/9) = A(M – K1h2 – K2h4-…..) + B(M-1/9K​1h2-1/81K2h4...)
+ C(M-1/81K1h2-1/6561K2h4-….)
= M(A+B+C) –K1h2(A+B/9+C/81) –...
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