Gaussian quadrature weights

Write a routine that computes the Gaussian quadrature weights and nodes f o, f i, , fjv ... 

In the case of 3b, Gaussian quadrature can be used, choosing the weighting function to remove the singularities from the desired integral. A variable step size differential equation integration routine [38, Chapter 15] produces the only practicable solution to 3c. 

Weights and zeros for the above formulas (and for other Gaussian formulas) may he found in references such as Stroud Gaussian Quadrature Formulas, Prentice-Hall, 1966). 

Two features of the trapezoidal method are that we use a uniform spacing between the positions where we evaluate fix) and that every evaluation of fix) (except the end points) is given equal weight. Neither of these conditions is necessary or even desirable. An elegant class of integration methods called Gaussian quadrature defines methods that have the form... 

It is important to note that these weight functions are exactly the weights of the Gaussian quadrature, this fact will be useful in the discussion of error bounds. 

It is well known that the determination of abscissas and weights of the Gaussian quadrature from power moments is an exponentially ill-conditioned problem due to the presence of rounding errors. 

Weights of Gaussian quadrature Length of ith finite element (fixed)... 

In Gaussian quadrature theory the NDF is called the weight function or measure. The weight function must be nonnegative and non-null in the integration interval and all its moments. 

Note that a different set of Laguerre polynomials, parameterized by k > 0, is used for each value of a. Thus, the weights and abscissas must be computed separately for each value of a. As with any Gaussian quadrature, Eq. (3.86) is exact when g(f) is a polynomial of order less than 2N (Gautschi, 2004). 

The definition of the Gaussian quadrature formula in Eq. E.30 implies that the determination of this formula is determined by the selection of N quadrature points and N quadrature weights that is, we have 2N parameters to be found. With these degrees of freedom (2N parameters), it is possible to fit a polynomial of degree 2N - 1. This means that if Xj and are properly chosen, the Gausssian quadrature formula can exactly integrate a polynomial of degree up to 2N — 1. 

We can also apply the Gaussian quadrature formula to the case when we have a weighting function in the integrand as... 

Table E.1 Quadrature Points and Weights for Gaussian Quadrature...

The points and weights of the Gaussian quadrature solve the moment equations for A = 0,1,..., 2 1 exactly as the pseudospqctrum of order n and the following... 

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