Applied Computational Economics and Finance by Mario J. Miranda

By Mario J. Miranda

Excellent booklet for utilized economics with many examples and usefull Matlab codes. first-class and invaluable Matlab toolkit.
However, the theoretical facet is comparatively vulnerable and never coated good.

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Example text

5 . 4. 3 Newton’s Method In practice, most nonlinear problems are solved using Newton’s method or one of its variants. Newton’s method is based on the principle of successive linearization. Successive linearization calls for a hard nonlinear problem to be replaced with a sequence of simpler linear problems whose solutions converge to the solution of the nonlinear problem. Newton’s method is typically formulated as a rootfinding technique, but may be used to solve a fixedpoint problem x = g(x) by recasting it as the rootfinding problem f (x) = x − g(x) = 0.

Interchanging the two rows at the outset of Gaussian elimination does not alter the theoretical solution to the linear equation, but allows one to perform Gaussian elimination with a diagonal element of larger magnitude. Consider the equivalent linear equation system after the rows have been interchanged: 1 −M −1 1 1 x1 x2 = 2 1 After interchanging the rows, the new A matrix may be factored as 1 −M −1 1 1 = 1 −M −1 0 1 1 0 1 M −1 + 1 Backward and forward substitution yield the theoretical results x1 = 1 − M −1 and x2 = M −1 + 1 + M −1 (1 − M −1 ).

Cholesky factorization requires only half as many operations as general Gaussian elimination and has the added advantage that it is less vulnerable to rounding error and does not require pivoting. The essential idea underlying Cholesky factorization is that any symmetric positive definite matrix A can be uniquely expressed as the product A=U U of an upper triangular matrix U and its transpose. The matrix U is called the Cholesky factor or square root of A. Given the Cholesky factor of A, the linear equation Ax = U U x = U (U x) = b may be solved efficiently by using forward substitution to solve U y=b and then using backward substitution to solve Ux = y.

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