Jacobi package examples

This package calculates a few orthogonal polynomials such as Jacobi polynomials $P_n^{\alpha,\beta}(x)$, Chebyshev polynomials of the first and second kinds, $T_n(x)$ and $U_n(x)$ and Legendre Polynomials $P_n(x) = P_n^{0,0}(x)$.

The polynomials can be calculated pointwise or a Poly structure can be obtained, which is basically made up of the polynomial coefficients.

The respective derivatives can also be calculated.

Pointwise calculation of polynomials



In [2]:

    
using Jacobi

To calculate the Jacobi polynomial $P_5^{0.2, 0.1}(0.3)$:



In [4]:

    
x = 0.3
a = 0.2
b = 0.1
m = 5
jacobi(x, m, a, b)









    Out[4]:





0.3780304524166406

The derivative $\left.\frac{dP_5^{0.2, 0.1}(x)}{dx}\right|_{x=0.3}$:



In [5]:

    
djacobi(x, m, a, b)









    Out[5]:





-0.4343764194492186

The Legendre, Chebyshev (first and second kind) are calculated as follows:



In [6]:

    
y1 = legendre(x, m) # Legendre polynomial of degree m
y2 = chebyshev(x, m) # Chebyshev polynomial of first kind of degree m
y3 = chebyshev2(x, m) # Chebyshev polynomial of second kind of degree m
[y1, y2, y3]









    Out[6]:





3-element Array{Float64,1}:
 0.345386
 0.99888 
 1.01376

Finally, the respective derivatives:



In [7]:

    
y1 = dlegendre(x, m) # Derivative of Legendre polynomial of degree m
y2 = dchebyshev(x, m) # Derivative of Chebyshev polynomial of first kind of degree m
y3 = dchebyshev2(x, m) # Derivative ofChebyshev polynomial of second kind of degree m
[y1, y2, y3]









    Out[7]:





3-element Array{Float64,1}:
 -0.168562
  0.248   
 -1.344

Polynomial coefficients

Sometimes it is useful to calculate the coefficients of the polynomials. The functions with prefix poly_ return a Poly object available in the package Polynomials.

The naming follows the pointwise calculation of polynomials already described above.



In [8]:

    
using Polynomials

T5 = poly_chebyshev(5, Int, :x)









    Out[8]:





Poly(5x - 20x^3 + 16x^5)



In [10]:

    
T8 = poly_chebyshev(8, Float64, :y)









    



Polynomials must have same variable
while loading In[10], in expression starting on line 1

 in * at /home/pjabardo/.julia/v0.3/Polynomials/src/Polynomials.jl:174
 in poly_chebyshev at /home/pjabardo/software/julia/Jacobi.jl/src/polynomials.jl:16



In [12]:

    
T4 = poly_chebyshev(8)









    Out[12]:





Poly(1 - 32x^2 + 160x^4 - 256x^6 + 128x^8)



In [13]:

    
dT4dx = poly_dchebyshev(4)









    Out[13]:





Poly(-16x + 32x^3)



In [14]:

    
U5 = poly_chebyshev2(5)









    Out[14]:





Poly(6x - 32x^3 + 32x^5)



In [15]:

    
dU5dx = poly_dchebyshev2(5)









    Out[15]:





Poly(6 - 96x^2 + 160x^4)



In [16]:

    
P6 = poly_legendre(6) # Legendre Polynomial









    Out[16]:





Poly(-0.3125 + 6.5625x^2 - 19.6875x^4 + 14.4375x^6)



In [17]:

    
dP6 = polyder(P6)# Derivative using Polynomials Package
dp6b = poly_dlegendre(6)  # Syntatic sugar calling the above line.









    Out[17]:





Poly(13.125x - 78.75x^3 + 86.625x^5)



In [18]:

    
err = dP6 - dp6b  # Both procedures above should be the same!









    Out[18]:





Poly(0.0)



In [19]:

    
P7ab = poly_jacobi(7, a, b) # Jacobi polynomials









    Out[19]:





Poly(-0.024746232255859372 - 2.4416546568652344x + 0.5967953141894532x^2 + 22.727281485986328x^3 - 1.9930645195800785x^4 - 51.4035174487207x^5 + 1.5947440682714844x^6 + 32.57834310897459x^7)



In [20]:

    
dP7abdx = poly_djacobi(7, a, b)









    Out[20]:





Poly(-2.4416546568652344 + 1.1935906283789064x + 68.18184445795899x^2 - 7.972258078320314x^3 - 257.0175872436035x^4 + 9.568464409628906x^5 + 228.04840176282215x^6)

Calculating the polynomials at several points

Often, it is necessary to calculate polynomials at several points. As syntatic sugar a few functions are available to make that easy:



In [21]:

    
using PyPlot # Seeing is believing ...









    



INFO: Loading help data...



In [22]:

    
x = linspace(-1.0, 1.0, 401);



In [24]:

    
y1 = chebyshev(x, 8);
plot(x, y1);

There is the mutating function:



In [27]:

    
y2 = zeros(x);
chebyshev!(x, 5, y2)
plot(x, y2)









    












    Out[27]:





1-element Array{Any,1}:
 PyObject <matplotlib.lines.Line2D object at 0x7f77b7133080>

The same thing is implemented for Jacobi and Legendre polynomials and their derivative:



In [30]:

    
y3 = jacobi(x, 4, -0.5, 0.5)
y4 = djacobi(x, 4, -0.5, 0.5)
plot(x, y3, "b-", x, y4, "r--")









    












    Out[30]:





2-element Array{Any,1}:
 PyObject <matplotlib.lines.Line2D object at 0x7f77b6f5fb00>
 PyObject <matplotlib.lines.Line2D object at 0x7f77b6f5fe48>

Zeros of polynomials

oThe functions starting with suffix _zero calculate the zeros of the polynomials.



In [39]:

    
x = linspace(-1, 1, 401);
y = chebyshev(x, 6);
z = chebyshev_zeros(6)
plot(x, y)
axhline(y=0.0, color="black")
[axvline(x=zz, color="red") for zz in z];

Again there are mutating versions of this function:



In [42]:

    
z = zeros(5)
jacobi_zeros!(5, a, b, z)
z









    Out[42]:





5-element Array{Float64,1}:
 -0.899519 
 -0.537495 
 -0.0136203
  0.514423 
  0.887668

Types other than `Float64`

Most times, calculations will be carried out using Float64 because that is the most useful datatype. Most functions can be used with other data types, including Float32, BigFloat and, except when calculating zeros of polynomials, Rational{T}.



In [43]:

    
jacobi(float32(0.5), 8, 0.5, 1.0)









    Out[43]:





-0.11098426440730691



In [44]:

    
chebyshev(23//47, 7)









    Out[44]:





214832820407//506623120463



In [45]:

    
djacobi(BigFloat(0.6), 10, 0.5, -0.5)









    Out[45]:





4.770638984160155424568808763332988903749347442526164591900368406181365041596503e+00 with 256 bits of precision



In [46]:

    
jacobi_zeros(7, 1, -1, BigFloat)









    Out[46]:





7-element Array{BigFloat,1}:
 -1e+00                                                                               
 -8.717401485096066153374457612206634381037806696769833549194152855738989940929645e-01
 -5.917001814331423021445107313979531899457009895173324967422595914701481800491344e-01
 -2.092992179024788687686572603453512552955454050866810988908466923221309236462841e-01
  2.092992179024788687686572603453512552955454050866810988908466923221309236462841e-01
  5.917001814331423021445107313979531899457009895173324967422595914701481800491344e-01
  8.717401485096066153374457612206634381037806696769833549194152855738989940929645e-01