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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 7 September 7, 2011 Prof. Kamesh Madduri REVIEW 2 Last class: Vector Norms, Matrix Norms, and Conditioning Properties of a vector norm Matrix norm Properties of a matrix norm Condition number Error bounds and residual 3 Systems of Linear Equations Covered on blackboard, corresponding slides from textbook follow. Also see additional notes for examples covered in class (Ill upload them tomorrow, after class). 4 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Solving Linear Systems To solve linear system, transform it into one whose solution is same but easier to compute What type of transformation of linear system leaves solution unchanged? We can premultiply (from left) both sides of linear system Ax = b by any nonsingular matrix M without affecting solution Solution to MAx = Mb is given by x = ( MA ) 1 Mb = A 1 M 1 Mb = A 1 b Michael T. Heath Scientific Computing 26 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Example: Permutations Permutation matrix P has one 1 in each row and column and zeros elsewhere, i.e., identity matrix with rows or columns permuted Note that P 1 = P T Premultiplying both sides of system by permutation matrix, PAx = Pb , reorders rows, but solution x is unchanged Postmultiplying A by permutation matrix, APx = b , reorders columns, which permutes components of original solution x = ( AP ) 1 b = P 1 A 1 b = P T ( A 1 b ) Michael T. Heath Scientific Computing 27 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Example: Diagonal Scaling Row scaling: premultiplying both sides of system by nonsingular diagonal matrix D , DAx = Db , multiplies each row of matrix and righthand side by corresponding diagonal entry of D , but solution x is unchanged Column scaling: postmultiplying A by D , ADx = b , multiplies each column of matrix by corresponding diagonal entry of D , which rescales original solution x = ( AD ) 1 b = D 1 A 1 b Michael T. Heath Scientific Computing 28 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Triangular Linear Systems What type of linear system is easy to solve?...
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