1. Oct 18, 2012

### Ramona79

Hello everyone,

I have a question:
I want to solve and plot the following function with Gauss-Chebyshev quadrature using Mathematica code:

$$F(t_k)=\frac{1}{N}\sum_{i=1}^N\left[\sum_{j=1}^m a_jT_j(s_i)\right]\frac{1}{s_i-t_k}$$
wehre
$$s_i=\cos (\pi \frac{2i-1}{2N})\quad \quad i=1...N$$
$$t_k=\cos (\pi \frac{k}{N})\quad \quad i=1...N-1$$

on a quick answer I am very grateful

thank you

2. Nov 4, 2012

### jaloux3

F(t)=\int_ \! (\frac{1}{\sqrt{1-s^{2} } } \frac{Phi(s)}{t-s} ) \, ds

(integrals are from -1 to +1)
this equation may be solved by the Gauss-Chebyshev integration formulae:
assume that Phi(s) can be appoximated by the fallowing truncated series:

Phi(s)= \sum\limits_{j=1}^m a_{j}T_{j}(s)

so that the integral now reads

\sum\limits_{j=1}^m a_{j} \int_ \! (\frac{1}{\sqrt{1-s^{2} } } )(\frac{T_{j}(s) }{t-s} ) \, ds |t|<1
and my task is to evaluate the unknown coefficients a_{j} . The integral may be evaluated through the relation :

for j=0 :

\int_\! (\frac{1}{\sqrt{1-s^{2} } } )(\frac{T_{j}(s) }{t-s} ) \, ds = 0

for j>0 :

\int_ \! (\frac{1}{\sqrt{1-s^{2} } } )(\frac{T_{j}(s) }{t-s} ) \, ds = U_{j-1}(t)

so that

F(t)=\sum\limits_{j=1}^m a_{j} U_{j-1}(t)

we next note the fallowing relation :

for j=0
\frac{1}{N}\sum\limits_{i=1}^N \frac{T_{j}(s_{i}) }{s_{i}-t_{k}) } = 0
for 0<j<N :

\frac{1}{N}\sum\limits_{i=1}^N \frac{T_{j}(s_{i}) }{s_{i}-t_{k}) } = U_{j-1}(t_{k} )

where the points are the N roots of T_{N}(s) and the points t_{k} are the N-1 roots of U_{N-1}(t) .

It follows that

F(t_{k})=\sum\limits_{j=1}^m a_{j} U_{j-1}(t_{k})=\frac{\pi }{N} \sum\limits_{i=1}^N [ \sum\limits_{j=1}^m a_{j} T_{j}(s_{i}) ] \frac{1}{s_{i} -t_{k} } = \frac{\pi }{N} \sum\limits_{i=1}^N \frac{Phi(s_{i} )}{s_{i} -t_{k} }

where the integration points are:

s_{i} = \cos(\pi \frac{2i-1}{2N}) i=1...N

t_{k} = \cos(\pi \frac{k}{N}) i=1...N-1

the weights (\frac{\pi }{N} ) .

für das Gleichungssystem mit mehreren variablen

F(t_{k})=\sum\limits_{j=1}^m a_{j} U_{j-1}(t_{k}).

wo

F(t_{k}) und U_{j-1}(t_{k}) bekannt

und

a_{j} unbekannt.

wie kann ich bitte dieses Gleichungssystem

a_{j} = U_{j-1}(t_{k}) \ F(t_{k})

in MatLAB lösen.

mir fehlt Code.

3. Nov 4, 2012

ohhh pardon,
i rewrite it

4. Nov 4, 2012

### jaloux3

F(t)=∫($\frac{1}{\sqrt{1-s^2}}(\frac{\phi(s)}{t-s})ds$

(integrals are from -1 to +1)
this equation may be solved by the Gauss-Chebyshev integration formulae:
assume that Phi(s) can be appoximated by the fallowing truncated series:

$\phi(s) = Ʃ^{m}_{j=1} a_{j} T_{j}(s)$

so that the integral now reads

Ʃ$^{m}_{j=1}$ $a_{j}$∫($\frac{1}{\sqrt{1-s^2}}(\frac{T_{j}(s)}{t-s})ds$ ; -1<t<+1

and my task is to evaluate the unknown coefficients $a_{j}$ . The integral may be evaluated through the relation :

for j=0 :
∫($\frac{1}{\sqrt{1-s^2}}(\frac{T_{j}(s)}{t-s})ds$ = 0

for j>0 :
∫($\frac{1}{\sqrt{1-s^2}}(\frac{T_{j}(s)}{t-s})ds$ = $U_{j-1}(t)$

so that

F(t)=$\sum^{m}_{j=1}$ $a_{j}$ $U_{j-1}(t)$

we next note the fallowing relation :

for j=0

$\frac{1}{N}$ $\Sigma^{i=1}_{N}$ ($\frac{T_{j}(s_{i})}{t_{k}-s_{i}}$) = 0

for 0<j<N :

$\frac{1}{N}$ $\Sigma^{i=1}_{N}$ ($\frac{T_{j}(s_{i})}{t_{k}-s_{i}}$) = $U_{j-1}(t)$

where the points $s_{i}$ are the N roots of $T_{N}(s)$ and the points $t_{k}$ are the N-1 roots of $U_{N-1}(t)$ . It follows that

F($t_{k}$) = $\sum^{m}_{j=1}$ $a_{j}$ $U_{j-1}(t_{k})$ = $\frac{\pi}{N} Ʃ^{N}_{i=1} [\Sigma^{m}_{j=1} a_{j} T_{j}(s_{i}) ]\frac{1}{s_{i}-t_{k}} = \frac{\pi}{N}\Sigma^{N}_{i=1} \frac{\phi (s_{i})}{s_{i}-t_{k}}$

where the integration points are:

$s_{i} = cos(\pi \frac{2i-1}{2N})$ i=1....N

$t_{k} = cos(\pi \frac{k}{N})$ k=1....N-1

the weights ($\frac{\pi}{N}$)

Note that the integration has been reduced to the sum and weights ($\frac{\pi}{N}$) and the integration points $s_{i}$ are the same as used as in the standard Gaussian quadrature formula.

Let's have a look at :

F($t_{k}$) = $\sum^{m}_{j=1}$ $a_{j}$ $U_{j-1}(t_{k})$

We assume that F($t_{k}$) and $U_{j-1}(t_{k})$ are given. That leads to m equations in case there are m different $t_{k}$. It's our task to evaluate the unknown coefficients $a_{j}$.

Therefor i must solve a linear equation m multiple variables ( the unknown coefficients $a_{j}$ ) in MatLAB .

can you help me to solve it in MatLAB.
i need a code in matlab.