Pivoting Perturbation Analysis Scaling and Equilibration Perturbation Analysis
Pivoting, Perturbation Analysis, Scaling and Equilibration
Perturbation Analysis
Perturbation in forcing vector b:
Perturbation in matrix A: Product of perturbation quantities (negligible)
Condition Number:
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Is determinant a good measure of matrix conditioning? 8
Scaling and Equilibration: ü It helps to reduce the truncation errors during computation. ü Helps to obtain a more accurate solution for moderately illconditioned matrix. ü Example: Consider the following set of equations ü Scale variable x 1 = 103 × x 1ʹ and multiply the second equation by 100. Resulting equation is:
Scaling ü Vector x is replaced by xʹ such that, x = Sxʹ. ü S is a diagonal matrix containing the scale factors! ü For the example problem: ü Ax = b becomes: Ax = ASxʹ = Aʹxʹ = b where, Aʹ = AS ü For the example problem: ü Scaling operation is equivalent to post-multiplication of the matrix A by a diagonal matrix S containing the scale factors on the diagonal
Equilibration
Example Problem
Pivoting, Scaling and Equilibration (Recap) ü Before starting the solution algorithm, take a look at the entries in A and decide on the scaling and equilibration factors. Construct matrices E and S. ü Transform the set of equation Ax = b to EASxʹ = Eb ü Solve the system of equation Aʹxʹ = bʹ for xʹ, where Aʹ = EAS and bʹ = Eb ü Compute: x = Sxʹ ü Gauss Elimination: perform partial pivoting at each step k ü For all other methods: perform full pivoting before the start of the algorithm to make the matrix diagonally dominant, as far as practicable! ü These steps will guarantee the best possible solution for all wellconditioned and mildly ill-conditioned matrices! ü However, none of these steps can transform an ill-conditioned matrix to a well-conditioned one.
Iterative Improvement by Direct Methods ü For moderately ill-conditioned matrices an approximate solution x to the set of equations Ax = b can be improved through iterations using direct methods. ü Compute: r = b - A x ü Recognize: r = b - A x + Ax - b ü Therefore: A(x - x ) = AΔx = r ü Compute: x = x + Δx ü The iteration sequence can be repeated until ǁ Δx ǁ ≤ ε
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Solution of System of Nonlinear Equations
System of Non-Linear Equations f(x) = 0 f is now a vector of functions: f = {f 1, f 2, … fn}T x is a vector of independent variables: x = {x 1, x 2, … xn}T ü Open methods: Fixed point, Newton-Raphson, Secant
Open Methods: Fixed Point •
Open Methods: Fixed Point •
Open Methods: Newton-Raphson •
Open Methods: Newton-Raphson •
Open Methods: Newton-Raphson •
Example Problem: Tutorial 3 Q 2 • f(x) = 0
Open Methods: Newton-Raphson •
Open Methods: Secant ü Jacobian of the Newton-Raphson method is evaluated numerically using difference approximation. ü Numerical methods for estimation of derivative of a function will be covered in detail later. ü Rest of the method is same.
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