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
    • Gaussian_Quad
  • 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]