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2. Consider the function f(x) = -21° +1. (a) Calculate the interpolating...
Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) ...
Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating polynomial determined by m equidistant interpolation points, (2) an interpolating polynomial determined by interpolation at the m zeros of the Chebyshev polynomial T_m(x), and (3) by interpolating by cubic splines instead of by a polynomial. Estimate the approximation error by evaluation max_i |f(z_i)-p(z_i)| for many points z_i on [-1,1]. For instance, you could use 10m points z_i. The cubic spline interpolant can be determined in...
1. Consider the polynonial Pl (z) of degree 4 interpolating the function f(x) sin(x) on the interval n/4,4 at the e...
1. Consider the polynonial Pl (z) of degree 4 interpolating the function f(x) sin(x) on the interval n/4,4 at the equidistant points r--r/4, xi =-r/8, x2 = 0, 3 π/8, and x4 = π/4. Estimate the maximum of the interpolation absolute error for x E [-r/4, π/4 , ie, give an upper bound for this absolute error maxsin(x) P(x) s? Remark: you are not asked to give the interpolation polynomial P(r).
1. Consider the polynonial Pl (z) of degree 4...
3. (30 points) Let f(x) = 1/x and data points Zo = 2, x,-3 and x2 = 4. Note that you can use the ...
3. (30 points) Let f(x) = 1/x and data points Zo = 2, x,-3 and x2 = 4. Note that you can use the abscissae to find the corresponding ordinates (a) (8 points) Find by hand the Lagrange form, the standard form, and the Newton form of the interpolating polynomial p2(x) of f(x) at the given points. State which is which! Then, expand out the Newton and Lagrange form to verify that they agree with the standard form of p2...
Quiz 5. Due Wednesday May 22, 2019. zo- e 1,e, 2-. Give the representation Consider interpolating the function In(x (without developing and 'simplifying in al) of the interpolation po...
Quiz 5. Due Wednesday May 22, 2019. zo- e 1,e, 2-. Give the representation Consider interpolating the function In(x (without developing and 'simplifying in al) of the interpolation polynomial Pa(z) expressed by Without calculating In(2) and P2(2) estimate the absolute error IIn(2) - P2(2)] s?
Quiz 5. Due Wednesday May 22, 2019. zo- e 1,e, 2-. Give the representation Consider interpolating the function In(x (without developing and 'simplifying in al) of the interpolation polynomial Pa(z) expressed by
Without calculating In(2)...
**********************matlab code please******************* 1. Interpolation error of polynomial fit Using 11 equi-dist...
**********************matlab code please*******************
1. Interpolation error of polynomial fit Using 11 equi-distributed points (10 equal segments) in the interval [-1 1], Using Newton's form find and plot the interpolating polynomial p(x) for the function f(x) -1/(125x2). Comment on the large discrepancies between p(x) and the function f(x) that the data came from Write down an expression for the error in the interpolating polynomial above? Which part of the expression is responsible for the large errors observed?
1. Interpolation error of...
class: numerical analysis I wish if it was written in block letter Sorry I can't read...
class: numerical analysis
I wish if it was written in block letter
Sorry I can't read cursive
= COS Problem 1: Recall that the Chebyshev nodes x4, x1,...,xy are determined on the interval (-1,1] as the zeros of Tn+1(x) = cos((n + 1) arccos(x)) and are given by 2j +10 Xj j = 0,1, ... 1 n+1 2 Consider now interpolating the function f(x) = 1/(1 + x2) on the interval (-5,5). We have seen in lecture that if equispaced...
Compute, using divided differences, the value of the piecewise cubic Her- mite interpolating polynomial at x = 11=10 giv...
Compute, using divided differences, the value of the piecewise
cubic Her-
mite interpolating polynomial at x = 11=10 given nodes at xi = i,
for i = 1; : : : ; 10,
and values and derivatives at the nodes from the function f(x) =
1=x.
Remember iterative formula for divided differences:
2. (25 pts) Compute, using divided differences, the value of the piecewise cubic Her mite interpolating polynomial at x-11/10 given nodes at ai-i, for i-1, , 10. and...
numerical methods 2+17), j = 0,1...... Problem 1: Recall that the Chebyshev nodes x0, 71,..., are...
numerical methods
2+17), j = 0,1...... Problem 1: Recall that the Chebyshev nodes x0, 71,..., are determined on the interval (-1,1) as the zeros of Tn+1(x) = cos((n +1) arccos(x)) and are given by 2j +17 X; = cos in +12 Consider now interpolating the function f(x) = 1/(1+22) on the interval (-5,5). We have seen in lecture that if equispaced nodes are used, the error grows unbound- edly as more points are used. The purpose of this problem is...
2. Graph the functions f(x)x(x 1)(x-2) ..(x- k) for k- 1,2,..,10. (These are examples of the poly...
2. Graph the functions f(x)x(x 1)(x-2) ..(x- k) for k- 1,2,..,10. (These are examples of the polynomials occurring in the error formula for polynomial interpolation.) We want to produce an evenly spaced table of values for the function f(x) sin(x) for x E [O,T/2] such that, with cubic interpolation, we can give the values of the function at any point in the interval with an error less than 5 10-12. That means finding a number n such that with h-/2n...
please answer with good handwriting Q1 (a) Given the function f(x)= x - 5x² - 2x...
please answer with good handwriting
Q1 (a) Given the function f(x)= x - 5x² - 2x +10. (1) Prove that there at least a root in the interval [1,3] by using Intermediate Value Theorem. (2 marks) (b) (i) Find the root of f(x) by using Bisection method. Iterate until i = 5. (8 marks) Prove the Lagrange interpolating polynomial of second degree for data of (0,1), (1,2) and (4,2) is P2(x) = -* x2 + x + 1. (5 marks)...
1. Consider the polynonial Pl (z) of degree 4 interpolating the function f(x) sin(x) on the interval n/4,4 at the equidistant points r--r/4, xi =-r/8, x2 = 0, 3 π/8, and x4 = π/4. Estimate the maximum of the interpolation absolute error for x E [-r/4, π/4 , ie, give an upper bound for this absolute error maxsin(x) P(x) s? Remark: you are not asked to give the interpolation polynomial P(r).
1. Consider the polynonial Pl (z) of degree 4...
3. (30 points) Let f(x) = 1/x and data points Zo = 2, x,-3 and x2 = 4. Note that you can use the abscissae to find the corresponding ordinates (a) (8 points) Find by hand the Lagrange form, the standard form, and the Newton form of the interpolating polynomial p2(x) of f(x) at the given points. State which is which! Then, expand out the Newton and Lagrange form to verify that they agree with the standard form of p2...
Quiz 5. Due Wednesday May 22, 2019. zo- e 1,e, 2-. Give the representation Consider interpolating the function In(x (without developing and 'simplifying in al) of the interpolation polynomial Pa(z) expressed by Without calculating In(2) and P2(2) estimate the absolute error IIn(2) - P2(2)] s?
Quiz 5. Due Wednesday May 22, 2019. zo- e 1,e, 2-. Give the representation Consider interpolating the function In(x (without developing and 'simplifying in al) of the interpolation polynomial Pa(z) expressed by
Without calculating In(2)...
**********************matlab code please*******************
1. Interpolation error of polynomial fit Using 11 equi-distributed points (10 equal segments) in the interval [-1 1], Using Newton's form find and plot the interpolating polynomial p(x) for the function f(x) -1/(125x2). Comment on the large discrepancies between p(x) and the function f(x) that the data came from Write down an expression for the error in the interpolating polynomial above? Which part of the expression is responsible for the large errors observed?
1. Interpolation error of...
class: numerical analysis
I wish if it was written in block letter
Sorry I can't read cursive
= COS Problem 1: Recall that the Chebyshev nodes x4, x1,...,xy are determined on the interval (-1,1] as the zeros of Tn+1(x) = cos((n + 1) arccos(x)) and are given by 2j +10 Xj j = 0,1, ... 1 n+1 2 Consider now interpolating the function f(x) = 1/(1 + x2) on the interval (-5,5). We have seen in lecture that if equispaced...
Compute, using divided differences, the value of the piecewise
cubic Her-
mite interpolating polynomial at x = 11=10 given nodes at xi = i,
for i = 1; : : : ; 10,
and values and derivatives at the nodes from the function f(x) =
1=x.
Remember iterative formula for divided differences:
2. (25 pts) Compute, using divided differences, the value of the piecewise cubic Her mite interpolating polynomial at x-11/10 given nodes at ai-i, for i-1, , 10. and...
numerical methods
2+17), j = 0,1...... Problem 1: Recall that the Chebyshev nodes x0, 71,..., are determined on the interval (-1,1) as the zeros of Tn+1(x) = cos((n +1) arccos(x)) and are given by 2j +17 X; = cos in +12 Consider now interpolating the function f(x) = 1/(1+22) on the interval (-5,5). We have seen in lecture that if equispaced nodes are used, the error grows unbound- edly as more points are used. The purpose of this problem is...
2. Graph the functions f(x)x(x 1)(x-2) ..(x- k) for k- 1,2,..,10. (These are examples of the polynomials occurring in the error formula for polynomial interpolation.) We want to produce an evenly spaced table of values for the function f(x) sin(x) for x E [O,T/2] such that, with cubic interpolation, we can give the values of the function at any point in the interval with an error less than 5 10-12. That means finding a number n such that with h-/2n...
please answer with good handwriting
Q1 (a) Given the function f(x)= x - 5x² - 2x +10. (1) Prove that there at least a root in the interval [1,3] by using Intermediate Value Theorem. (2 marks) (b) (i) Find the root of f(x) by using Bisection method. Iterate until i = 5. (8 marks) Prove the Lagrange interpolating polynomial of second degree for data of (0,1), (1,2) and (4,2) is P2(x) = -* x2 + x + 1. (5 marks)...
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