Q&A for scientists using computers to solve scientific problems. By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems.

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Romberg integration is a process used to compute finite integrals requiring very high accuracy (super‑low error). This integration process, first proposed by Werner Romberg in 1955, is remarkable because it employs a series smaller‑sized less‑accurate integral approximations, typically obtained...

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Since , , , and are all known functions, we can evaluate these linear equations at collocation nodes inside the integration range. This yields linear equations for the coefficients . Upon solving these equations numerically, we have an approximate form for the , and we may directly evaluate the integral using the fundamental theorem of calculus, as

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Use Taylor Series to approximate functions and evaluate the approximations ... Numerical Integration 1.5 weeks Trapezoid rule (Sec. 21.1), Romberg algorithm (Sec 22.2 ...

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11. Write down the three point Gaussian quadrature formula to evaluate . 12. State the Romberg’s integration formula with h 1 and h 2 further obtain the formula when h 1 = h and h 2 = h/2. 13. Use two point Gaussian quadrature formula to solve 14. Evaluate dx by two point Gaussian quadrature formula. 15.

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352x 17y 4z using Gauss elimination method. Or (b) Using Gauss-Jordan method, solve the following equations. 42x1 x2 4x3 ; 5x1 3x2 x3 ; 13x1 2x2 2x3 . 5. (a) Evaluate 1 0 exdx by Simpson’s one-third rule correct five decimal places. Or (b) Evaluate 1 0 1 x2 dx by using Romberg’s method.