4. The following algorithm for approximating roots combines the bisection method with Newton's method and ensures the convergence even in the case where Newton's method fail to converge. Choose an interval [a,b] such that f(a)f(b) ≤0. Compute initial approximation xo = (a+b)/2 of the root using the bisection method. For i=1,2,... Compute i using Newton's method and xi-1. If xi [a,b], set x₁ = (a + b)/2 (from bisection). Check for convergence. If f(a) f(xi) ≤0 set b = xi, else set a = xį . (a) Implement this algorithm in a PYTHON function with the following specifications: def findzero (a, b, tol, maxit, f,df) # Input: # a, b = The endpoints of the interval # tol = The required tolerance # maxit = Maximum number of iterations # f, df = The function and its derivative # Output: # star = approximation of root # niter = number of iterations for convergence #ierr = # # # 0, the method converged 1, df was zero or undefined 2, maximum number of iterations has been reached return xstar, niter, ierr
4. The following algorithm for approximating roots combines the bisection method with Newton's method and ensures the convergence even in the case where Newton's method fail to converge. Choose an interval [a,b] such that f(a)f(b) ≤0. Compute initial approximation xo = (a+b)/2 of the root using the bisection method. For i=1,2,... Compute i using Newton's method and xi-1. If xi [a,b], set x₁ = (a + b)/2 (from bisection). Check for convergence. If f(a) f(xi) ≤0 set b = xi, else set a = xį . (a) Implement this algorithm in a PYTHON function with the following specifications: def findzero (a, b, tol, maxit, f,df) # Input: # a, b = The endpoints of the interval # tol = The required tolerance # maxit = Maximum number of iterations # f, df = The function and its derivative # Output: # star = approximation of root # niter = number of iterations for convergence #ierr = # # # 0, the method converged 1, df was zero or undefined 2, maximum number of iterations has been reached return xstar, niter, ierr
Computer Networking: A Top-Down Approach (7th Edition)
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![4.
The following algorithm for approximating roots combines the bisection method with
Newton's method and ensures the convergence even in the case where Newton's method fail to
converge.
Choose an interval [a, b] such that f(a)f(b) ≤ 0.
Compute initial approximation xo = (a+b)/2 of the root using the bisection method.
For i=1,2,...
Compute x₁ using Newton's method and X₁-1.
If x₁ [a, b], set x₁ = (a + b)/2 (from bisection).
Check for convergence.
If f(a) f(xi) ≤0 set b = xi, else set a = xį.
(a)
Implement this algorithm in a PYTHON function with the following specifications:
def findzero (a, b, tol, maxit, f, df)
# Input:
# a, b = The endpoints of the interval
# tol = The required tolerance
# maxit = Maximum number of iterations
# f, df = The function and its derivative
# Output:
# star
#niter
# ierr
#
#
#
approximation of root
= number of iterations for convergence
0, the method converged
1, df was zero or undefined
2, maximum number of iterations has been reached
return xstar, niter, ierr](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01b4eff1-1c92-4d86-9563-264f29556131%2Fcfcf9fe5-e240-4f9a-b83d-ad271309b8cc%2F88pmr7o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.
The following algorithm for approximating roots combines the bisection method with
Newton's method and ensures the convergence even in the case where Newton's method fail to
converge.
Choose an interval [a, b] such that f(a)f(b) ≤ 0.
Compute initial approximation xo = (a+b)/2 of the root using the bisection method.
For i=1,2,...
Compute x₁ using Newton's method and X₁-1.
If x₁ [a, b], set x₁ = (a + b)/2 (from bisection).
Check for convergence.
If f(a) f(xi) ≤0 set b = xi, else set a = xį.
(a)
Implement this algorithm in a PYTHON function with the following specifications:
def findzero (a, b, tol, maxit, f, df)
# Input:
# a, b = The endpoints of the interval
# tol = The required tolerance
# maxit = Maximum number of iterations
# f, df = The function and its derivative
# Output:
# star
#niter
# ierr
#
#
#
approximation of root
= number of iterations for convergence
0, the method converged
1, df was zero or undefined
2, maximum number of iterations has been reached
return xstar, niter, ierr
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