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Linear Algebra
A natural cubic spline for a function f(x) is defined by ſax° +15x² –...
2. Consider the cubic spline for a function f on [0, 2] defined by S(x) =...
2. Consider the cubic spline for a function f on [0, 2] defined by S(x) = { ={ (z. 2x3 + ax2 + rx +1 if 0 < x <1 (x - 1)3 + c(x - 1)2 + d(x - 1) + ß if 1 < x < 2 where r, c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
2. A natural cubic spline S is defined by S(z) = { $(") = 11 S....
2. A natural cubic spline S is defined by S(z) = { $(") = 11 S. (C)=1+ B(-1) + D(x - 1)3 S (2)=1+b(-2) + 3(x - 2)2 + (x - 2) if 1 <3 <2 if 2 <3<3 Find B, D, b, and d. Hint: Recall that a natural cubic spline has S" equal to 0 at the endpoints.
beta is 1 2. ( 20p.) Consider the cubic spline for a function f on (0,...
beta is 1
2. ( 20p.) Consider the cubic spline for a function f on (0, 2) defined by 223 + ax2 + rx + 1 if 0 < x < 1 S(X) = (.x - 1)3 + c(x - 1)2 + d(x - 1) + B if i < x < 2 S(x) = { where r, c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
Alpha=9 beta=3 yazarsin 2. ( 20p.) Consider the cubic spline for a function f on (0,2)...
Alpha=9 beta=3 yazarsin
2. ( 20p.) Consider the cubic spline for a function f on (0,2) defined by 2x3 + x² +rx +1 if 0 <x<1 S(x) = (x - 1)3 + c(x - 1)2 + d(x - 1) + B if 1<x<2 = {(2-1) where r,c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
5. (a) The natural spline S(a) passing through the n+ points is a collection of n cubic functions S,(x) defined in the n intervals x, Sxx Suppose that all the points are equally spaced, with unifo...
5. (a) The natural spline S(a) passing through the n+ points is a collection of n cubic functions S,(x) defined in the n intervals x, Sxx Suppose that all the points are equally spaced, with uniform point spacing h=5m-x, for jso,1,..,,n. Ifthe symbols M,, 0, represent the second derivatives of the spline at cach of the mesh points, show that in each intervl-.R-1 For the natural cubic spline (for which M, O and M-=0 ), show that the moments M...
Find the natural cubic spline interpolant to f(x) = e^(x^3) at the nodes {xi} i=0 = {−2, 0, 2} . ...
Find the natural cubic spline interpolant to f(x) = e^(x^3) at the nodes {xi} i=0 = {−2, 0, 2} . Calculate the value of the interpolant at x = 1. What is the error at this point?
QUESTION 4 (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790,...
QUESTION 4 (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790, F (3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system x2 + y = 9, 22 + y2 = 25, x, y > 0. Perform one iteration of Newton's method to approximate the solution, starting with (2, 4) as...
Homework Problem Set 16 x-1)3 for 0 sx$1 a cubic spline? 1. Is the function s (x) = {2(-1)3 for 1...
Homework Problem Set 16 x-1)3 for 0 sx$1 a cubic spline? 1. Is the function s (x) = {2(-1)3 for 1 < x 2 -5 +8x-6x2 2x3 for1 sx s 2 Is the function s(x)-[ natural? Let x for j 1,2,3,4 by a calculation on paper 3 a cubic spline? Is it [1,0,1,0]. Determine the natural spline s with s(y)-yi 2. 18x2-2x3 for 2
Natural Cubic spline If the and Derivatives at the endpoints are known, ine. Pi" (x)= Y,...
Natural Cubic spline If the and Derivatives at the endpoints are known, ine. Pi" (x)= Y, 4 and Puni (xu) z Yu" for some numbers y," and ya theni 25, + z = 3y[x, 1x23 - į sxy," (al Sunit zou z 3y [xn-1 , xu] + & Sx uni Yu" (6) 10 Derive equations (a) and (b) for the natural cubic Spline
QUESTION 4 (10) (a) A natural cubic spline that fits the data given by f(3.0) =...
QUESTION 4 (10) (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790, f(3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system (10) 2+y=9, 22 + y2 = 25, 2,y> 0. Perform one iteration of Newton's method to approximate the solution, starting with (2, 4) as the initial solution. [20]
2. Consider the cubic spline for a function f on [0, 2] defined by S(x) = { ={ (z. 2x3 + ax2 + rx +1 if 0 < x <1 (x - 1)3 + c(x - 1)2 + d(x - 1) + ß if 1 < x < 2 where r, c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
2. A natural cubic spline S is defined by S(z) = { $(") = 11 S. (C)=1+ B(-1) + D(x - 1)3 S (2)=1+b(-2) + 3(x - 2)2 + (x - 2) if 1 <3 <2 if 2 <3<3 Find B, D, b, and d. Hint: Recall that a natural cubic spline has S" equal to 0 at the endpoints.
beta is 1
2. ( 20p.) Consider the cubic spline for a function f on (0, 2) defined by 223 + ax2 + rx + 1 if 0 < x < 1 S(X) = (.x - 1)3 + c(x - 1)2 + d(x - 1) + B if i < x < 2 S(x) = { where r, c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
Alpha=9 beta=3 yazarsin
2. ( 20p.) Consider the cubic spline for a function f on (0,2) defined by 2x3 + x² +rx +1 if 0 <x<1 S(x) = (x - 1)3 + c(x - 1)2 + d(x - 1) + B if 1<x<2 = {(2-1) where r,c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
5. (a) The natural spline S(a) passing through the n+ points is a collection of n cubic functions S,(x) defined in the n intervals x, Sxx Suppose that all the points are equally spaced, with uniform point spacing h=5m-x, for jso,1,..,,n. Ifthe symbols M,, 0, represent the second derivatives of the spline at cach of the mesh points, show that in each intervl-.R-1 For the natural cubic spline (for which M, O and M-=0 ), show that the moments M...
QUESTION 4 (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790, F (3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system x2 + y = 9, 22 + y2 = 25, x, y > 0. Perform one iteration of Newton's method to approximate the solution, starting with (2, 4) as...
Homework Problem Set 16 x-1)3 for 0 sx$1 a cubic spline? 1. Is the function s (x) = {2(-1)3 for 1 < x 2 -5 +8x-6x2 2x3 for1 sx s 2 Is the function s(x)-[ natural? Let x for j 1,2,3,4 by a calculation on paper 3 a cubic spline? Is it [1,0,1,0]. Determine the natural spline s with s(y)-yi 2. 18x2-2x3 for 2
Natural Cubic spline If the and Derivatives at the endpoints are known, ine. Pi" (x)= Y, 4 and Puni (xu) z Yu" for some numbers y," and ya theni 25, + z = 3y[x, 1x23 - į sxy," (al Sunit zou z 3y [xn-1 , xu] + & Sx uni Yu" (6) 10 Derive equations (a) and (b) for the natural cubic Spline
QUESTION 4 (10) (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790, f(3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system (10) 2+y=9, 22 + y2 = 25, 2,y> 0. Perform one iteration of Newton's method to approximate the solution, starting with (2, 4) as the initial solution. [20]
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