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Question 1 2 pts The Hermite Interpolation polynomial for 33 distinct nodes has Degree at most...
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...
Let f(x) = xlnx. Approximate f(2) by the Hermite interpolating polynomial using x0 = 1 and...
Let f(x) = xlnx. Approximate f(2) by the Hermite interpolating polynomial using x0 = 1 and x1 = e and compare the error.(e ≈ 2.7...)
with distinct nodes, prove there is at most one polynomial of degree ≤ 2n + 1 that interpolates the data. Remember the F...
with distinct nodes, prove there is at most one polynomial of
degree ≤ 2n + 1 that interpolates the data. Remember the
Fundamental Theorem of Algebra says a nonzero polynomial has number
of roots ≤ its degree. Also, Generalized Rolle’s Theorem says if r0
≤ r1 ≤ . . . ≤ rm are roots of g ∈ C m[r0, rm], then there exists ξ
∈ (r0, rm) such that g (m) (ξ) = 0.
1. (25 pts) Given the table...
please answer question 2 only, question1 is the information that might need for question 2 2. Define the divided diffe...
please answer question 2 only, question1 is the information that
might need for question 2
2. Define the divided difference f[xo,xi,'. . ,Tk] as the coefficient of rk in p in Q.1. Prove the following recurrence formula: f(ax1, 2,,X- f{X0, X1,**.,&k-1 f[xo, ,,Xk] 1. Let f a, b -» IR and ro, x1, , Tk be k + 1 distinct points in [a, b]. Show that there exists a unique polynomial pk of degree <k such that ph (xj)f(x), j...
This assignment is about polynomial interpolation. 1) The user should be able to enter: a. A...
This assignment is about polynomial interpolation. 1) The user should be able to enter: a. A function named f. b. A number of points (nodes) with their respective values. c. A point x0 2) The output should be: a. A Newton Divided Differences polynomial (function of x) that approximates the function with agreement in the points. b. An approximation of f(x0) by Newton Divided Differences polynomial. c. The approximation absolute and relative errors.
er Lagrange ,Divided difference and Hermitewatnejed, Jnp 1.5, and x2-2, andf (x)ssin(x) * Given the point sx.-1, a) Find its Lagrange interpolation P on these points b) Write its newton's divi...
er Lagrange ,Divided difference and Hermitewatnejed, Jnp 1.5, and x2-2, andf (x)ssin(x) * Given the point sx.-1, a) Find its Lagrange interpolation P on these points b) Write its newton's divided difference P, polynomial c)Write Hermite Hs by Using part a outcomes d) Write Hermite Hi by Using part b outcomes Rules: Lagrange form of Hermite polynomial of degre at most 2n-+1 Here, L., (r) denotes the Lagrange coefficient polynomial of degree n. If ec la.bl, then the error formula...
Incorrect Question 5 0/2 pts What is the degree of the polynomial obtained by cubic spline...
Incorrect Question 5 0/2 pts What is the degree of the polynomial obtained by cubic spline interpolation function? It will be 3. Maximum 3. Minimum 3. It will be piecewise and the highest degree of each piece (between two consecutive points)is 3. It will be piecewise and the degree of each piece (between two consecutive points) is 3. If the number of points are more than 3, its degree will be 3.
Let (xi , f(xi)), i = 0, . . . , 3, be data points, where xi = i + 2, for i = 0, . . . , 3. Given the divided difference...
Let (xi , f(xi)), i = 0, . . . , 3, be data points, where xi = i
+ 2, for i = 0, . . . , 3. Given the divided differences f[x0] = 1,
f[x0, x1] = 2, f[x0, x1, x2] = −7, f[x0, x1, x2, x3] = 9, add the
data point (0, 3), find a Newton form for the Lagrange polynomial
interpolating all 5 data points.
3. (25 pts) Let (r,, f()), 0,3, be data...
If the polynomial x3 + x2 – 2 is divided by x + 1, the remainder...
If the polynomial x3 + x2 – 2 is divided by x + 1, the remainder is 0. True O False
Question 13 3 pts Let fe C41–3, 6] and suppose max_35x56 \54(x)| 5.87 and consider the...
Question 13 3 pts Let fe C41–3, 6] and suppose max_35x56 \54(x)| 5.87 and consider the nodes -3,-2,0, 2, 6. Let S(x) be the unique Clamped Spline interpolant to f(x) on these nodes. Then [f(x) – S(x)is bounded above by (give an answer correct to 2 decimal places)
with distinct nodes, prove there is at most one polynomial of
degree ≤ 2n + 1 that interpolates the data. Remember the
Fundamental Theorem of Algebra says a nonzero polynomial has number
of roots ≤ its degree. Also, Generalized Rolle’s Theorem says if r0
≤ r1 ≤ . . . ≤ rm are roots of g ∈ C m[r0, rm], then there exists ξ
∈ (r0, rm) such that g (m) (ξ) = 0.
1. (25 pts) Given the table...
please answer question 2 only, question1 is the information that
might need for question 2
2. Define the divided difference f[xo,xi,'. . ,Tk] as the coefficient of rk in p in Q.1. Prove the following recurrence formula: f(ax1, 2,,X- f{X0, X1,**.,&k-1 f[xo, ,,Xk] 1. Let f a, b -» IR and ro, x1, , Tk be k + 1 distinct points in [a, b]. Show that there exists a unique polynomial pk of degree <k such that ph (xj)f(x), j...
er Lagrange ,Divided difference and Hermitewatnejed, Jnp 1.5, and x2-2, andf (x)ssin(x) * Given the point sx.-1, a) Find its Lagrange interpolation P on these points b) Write its newton's divided difference P, polynomial c)Write Hermite Hs by Using part a outcomes d) Write Hermite Hi by Using part b outcomes Rules: Lagrange form of Hermite polynomial of degre at most 2n-+1 Here, L., (r) denotes the Lagrange coefficient polynomial of degree n. If ec la.bl, then the error formula...
Incorrect Question 5 0/2 pts What is the degree of the polynomial obtained by cubic spline interpolation function? It will be 3. Maximum 3. Minimum 3. It will be piecewise and the highest degree of each piece (between two consecutive points)is 3. It will be piecewise and the degree of each piece (between two consecutive points) is 3. If the number of points are more than 3, its degree will be 3.
Let (xi , f(xi)), i = 0, . . . , 3, be data points, where xi = i
+ 2, for i = 0, . . . , 3. Given the divided differences f[x0] = 1,
f[x0, x1] = 2, f[x0, x1, x2] = −7, f[x0, x1, x2, x3] = 9, add the
data point (0, 3), find a Newton form for the Lagrange polynomial
interpolating all 5 data points.
3. (25 pts) Let (r,, f()), 0,3, be data...
If the polynomial x3 + x2 – 2 is divided by x + 1, the remainder is 0. True O False
Question 13 3 pts Let fe C41–3, 6] and suppose max_35x56 \54(x)| 5.87 and consider the nodes -3,-2,0, 2, 6. Let S(x) be the unique Clamped Spline interpolant to f(x) on these nodes. Then [f(x) – S(x)is bounded above by (give an answer correct to 2 decimal places)
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