# 统计代写|主成分分析代写Principal Component Analysis代考|Title: Efficient Numerical Integration with Gaussian Quadrature

The passage discusses the Gaussian quadrature method for numerical integration, particularly in the context of finite element analysis. It explains that while exact integration of polynomial functions is possible analytically, it can be very cumbersome for complex integrands or when dealing with numerous integrations needed for large-scale finite element models. Gaussian quadrature provides an efficient way to approximate integrals by selecting a finite number of sampling points and associated weighting factors that guarantee exact integration of polynomials up to a certain degree.

The core idea is that for an integral of the form:

I=∫−11h(r) drI = \int_{-1}^{1} h(r) \, drI=∫ −1 1 ​ h(r)dr

When employing Gaussian quadrature with mmm sampling points, the integral can be approximated as:

I≈∑i=1mWih(ri)I \approx \sum_{i=1}^{m} W_i h(r_i)I≈∑ i=1 m ​ W i ​ h(r i ​ )

Where WiW_iW i ​ are the Gaussian weighting factors and rir_ir i ​ are the sampling points. The number of points mmm and their locations are chosen so that for a polynomial of order 2m−12m – 12m−1, the integral will be computed exactly.

For instance, a cubic polynomial (order n=3n = 3n=3) requires m=2m = 2m=2 sampling points. Using the conditions derived from integrating the polynomial’s coefficients, it is found that for a cubic polynomial, suitable points and weights are r1=−3/3r_1 = -\sqrt{3}/3r 1 ​ =− 3 ​ /3, r2=3/3r_2 = \sqrt{3}/3r 2 ​ = 3 ​ /3, and W1=W2=1W_1 = W_2 = 1W 1 ​ =W 2 ​ =1.

In the specific example provided, the integral

I=∫−11(r2−3r+7) drI = \int_{-1}^{1} (r^2 – 3r + 7) \, drI=∫ −1 1 ​ (r 2 −3r+7)dr

can be evaluated exactly using Gaussian quadrature with two sampling points and weights as listed above. The calculation results in I=14.666667I = 14.666667I=14.666667, which matches the exact answer obtained through analytical integration.

Moreover, Gaussian quadrature can be extended to multiple dimensions by performing the integration sequentially for each dimension. For a two-dimensional integral like

I=∫−11∫−11f(r,s) dr dsI = \int_{-1}^{1} \int_{-1}^{1} f(r,s) \, dr \, dsI=∫ −1 1 ​ ∫ −1 1 ​ f(r,s)drds

one applies the Gaussian quadrature rule twice – first for the integral with respect to rrr, and then for the remaining integral with respect to sss. The number of sampling points in each dimension is determined by the highest polynomial order present in the integrand.

The text also provides an example of a two-dimensional integral and demonstrates how to calculate its exact value using Gaussian quadrature. The result is confirmed to match the exact solution, showing the power of this numerical integration technique for finite element analysis and other applications involving polynomial integrands.

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