# Numerical Analysis

It's a julia package for Numerical Analysis.

Now, i just write a little function in it. I am a beginner of Julia and numerical analysis, there may be many problems, 👏 to discuss with me.🤣

## Basic

here are some function:

Now, i just write some functions:

• Basic
• N the derivative, use ForwardDiff packageto calculate $\frac{dy}{dx}$, then recursive to get Nth derivative.(emmm, I feel a bit slow)
• Taylor Polynomial, get the value nth Taylor Ploynomial.
• Solutions for equation in one Variable
• Bisection function, find root
• fixed_point function.
• Newton's Method
• The Secant Method
• The False Position Method
• Modified Newton's Method
• Müller’s Method
• Interpolation and the Lagrange Polynomial
• nth Larange interpolating polynomial
• Neville’s Iterated Interpolation
• Newton’s Divided-Difference Formula
• Natural Cubic Spline
• Clamped Cubic Spline
• Numerical Differentiation and integration
• Differentiation
• Three-Point and Five-Point formula
• Integration
• Trapezoidal Rule
• Simpson's Rule
• Newton_Cotes
• Romberg
• Mutiple Integrals
• SimpsonDoubleIntegral
• GaussianDoubleIntegral

• NthDerivative(f, x, n)
using NumericalAnalysis
NthDerivative(x->x^3, 4, 4) # 0
0
NthDerivative(x->x^3, 4, 2) # 24
24
NthDerivative(cos, 1, 5) # NthDerivative(cos, 1, 5)
-0.8414709848078965

• TaylorPolynomials(f, x, x₀, n)
TaylorPolynomials(cos, 0.1, 0, 6) # TaylorPolynomials(cos, 0.1, 0, 6)
0.9950041652777778
TaylorPolynomials(x->x^3, 1.1, 1, 6) #1.331000..
1.3310000000000004

• Bisection(f, a, b)
using NumericalAnalysis: SEq1
SEq1.Bisection(sin, π/2, 3π/2)
3.14159274721655
SEq1.Bisection(x->x-1, 0, 3π/2, output_seq=true)
26-element Array{Any,1}:
[2.356194490192345, 1.3561944901923448]
[1.1780972450961724, 0.17809724509617242]
[0.5890486225480862, -0.4109513774519138]
[0.8835729338221293, -0.11642706617787069]
[1.030835089459151, 0.03083508945915092]
[0.9572040116406402, -0.04279598835935983]
[0.9940195505498955, -0.0059804494501044525]
[1.0124273200045233, 0.01242732000452329]
[1.0032234352772094, 0.0032234352772093633]
[0.9986214929135524, -0.0013785070864476001]
⋮
[1.0000056708901213, 5.670890121267647e-6]
[0.9999966827214422, -3.3172785578461372e-6]
[1.0000011768057817, 1.1768057817107547e-6]
[0.9999989297636119, -1.0702363880676913e-6]
[1.0000000532846969, 5.328469687704285e-8]
[0.9999994915241543, -5.084758456508354e-7]
[0.9999997724044256, -2.2759557438689626e-7]
[0.9999999128445612, -8.71554387549267e-8]
[0.9999999830646291, -1.6935370883430778e-8]