lucafe-PCE4UDDE_matlab_codes.zip

# Orthogonal Polynomials and Approximation Theory In this folder, you can find the codes that are used in the third chapter of the Master thesis: *“Orthogonal Polynomials and Approximation Theory”*. In particular, you can find: - **figure\_hermiteH** [*script*] plots the first five Physicist Hermite orthogonal polynomials in the interval [-2,2]. The Matlab function **hermiteH** is used, hence to run this script Matlab r2014b is needed at least. *Figure 3.2 was realized using this script.* - **figure\_hermiteHe** [*script*] plots the first five Probabilistic Hermite orthogonal polynomials in the interval [-2,2], highlighting their zeros. Running this script requires that the functions **hermite** and **GaussHermite** are in the same folder. *Figure 3.3 was realized using this script.* - **figure\_laguerre** [*script*] plots the first five Laguerre orthogonal polynomials in the interval [0,5], highlighting their zeros. Running this script requires that the functions **laguerre** and **GaussLaguerre** are in the same folder. *Figure 3.6 was realized using this script.* - **figure\_legendre** [*script*] plots the first five Legendre orthogonal polynomials in the interval [-1,1], highlighting their zeros. Running this script requires that the functions **legendre** and **GaussLagendre** are in the same folder. *Figure 3.4 was realized using this script.* - **figure\_legendre\_sh** [*script*] plots the first five shifted Legendre orthogonal polynomials in the interval [0,1], highlighting their zeros. Running this script requires that the functions **legendre_sh** and **GaussLagendre_sh** are in the same folder. *Figure 3.5 was realized using this script.* - **figure\_pochhammer\_symbol** [*script*] plots the first seven Pochhammer functions in the interval [-4,4]. The Matlab function **pochhammer** is used, hence to run this script Matlab r2014b is needed at least. *Figure 3.1 was realized using this script.* - **[zeros,weight] = GaussHermite(n,sum\_weight)** [*function*] evaluates the nodes **zeros** and associated **weight** for the Gauss Hermite Quadrature Rule. It requires in input the number **n** of nodes that the users want with the associated weights. **n** is the degree of the probabilistic Hermite polynomial and its zeros are the corresponding nodes. It is possible to specify also the value of the sum of all the weights, **sum\_weight** (for default this value is set equal to 1). - **[zeros,weight] = GaussLaguerre(n,sum\_weight)** [*function*] evaluates the nodes ** zeros** and associated weights **weight** for the Gauss Laguerre Quadrature Rule. It requires in input the number **n** of nodes that the users want with the associated weights. **n** is the degree of the Laguerre polynomial and its zeros are the corresponding nodes. It is possible to specify also the value of the sum of all the weights **sum\_weight** (for default this value is set equal to 1). - **[zeros,weight] = GaussLegendre(n,sum\_weight)** [*function*] evaluates the nodes **zeros** and associated weights **weight** for the Gauss Legendre Quadrature Rule. It requires in input the number **n** of nodes that the users want with the associated weights. **n** is the degree of the Legendre polynomial and its zeros are the corresponding nodes. It is possible to specify also the value of the sum of all the weights **sum\_weight** (for default this value is set equal to 1). - **[zeros, weight] = GaussLegendre\_changeinterval(zeros,weight,a,b)** [*function*] translates the nodes **zeros** and **weight** generated by the function **GaussLegendre** in a opportune interval [**a**,**b**]. If the interval [**a**,**b**]is not specified, the program for default set [a,b]=[0,1]. It is not required any extra function to run this function. *Please refer to equation (3.59)* - **[zeros,weight] = GaussLegendre\_sh(n)** [*function*] evaluates the nodes **zeros** and associated weights **weight** for the Gauss shifted Legendre Quadrature Rule. It requires in input the number **n** of nodes that the users want with the associated weights. **n** is the degree of the shifted Legendre polynomial and its zeros are the corresponding nodes. This function merges efficiently the function **GaussLegendre** and **GaussLegendre\_changeinterval**, in any case it is not required any extra function to run this function. - **[h]=hermite(n,x)** [*function*] computes the probabilistic Hermite polynomials of degree **n**. **x** is the optional values where we want to evaluate the probabilistic Hermite polynomial of degree **n**. **x** can be a scalar a vector or a matrix. If **x** is omitted then **h** is an array with (**n**+1) elements that contains coefficients of each probabilistic Hermite polynomial term. If **x** is given, then **h** = He\_ **n** (**x**) and **h** has the same size of **x**. - **[l]=laguerre(n,x)** [*function*] computes the Laguerre polynomials of degree **n**. **x** is the optional values where we want to evaluate the Laguerre polynomial of degree **n**. If **x** is omitted then **l** is an array with (**n**+1) elements that contains coefficients of each Laguerre polynomial term. If **x** is given, then **l** = L\_ **n** (**x**) and **l** has the same size of **x**. - **[p]=legendre(n,x)** [*function*] computes the Legendre polynomials of degree **n**. **x** is the optional values where we want to evaluate the Legendre polynomial of degree **n**. If **x** is omitted then **p** is an array with (**n**+1) elements that contains coefficients of each Legendre polynomial term. If **x** is given, then **p** = P\_ **n** (**x**) and **p** has the same size of **x**. - **[p]=legendre\_sh(n,x)** [*function*] computes the shifted Legendre polynomials of degree **n**. **x** is the optional values where we want to evaluate the shifted Legendre polynomial of degree **n**. If **x** is omitted then **p** is an array with (**n** +1) elements that contains coefficients of each shifted Legendre polynomial term. If **x** is given, then **p** = P^\* \_ **n** (**x**) and **p** has the same size of **x**. The development of the algorithms for the calculation of orthogonal polynomials is based on the efficient algorithm of Avan Suinesiaputra and Fadillah Z Tala for computing physicists Hermite polynomials [hermite(n,x) of File Exchange](http://it.mathworks.com/matlabcentral/fileexchange/27746-hermite-polynomials/content/hermite.m) [Return to the main folder](https://github.com/lucafe/PCE4UDDE_matlab_codes).