NumericalMethodsChaper02

NumericalMethodsChaper02 - 1. Numerical Methods - Chapter 2...

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1 1. Numerical Methods - Chapter 2 1.1. Vector Norms p n i p i x x / 1 1 = = p Name Formula 1 1-norm Cartesian norm Manhattan norm = = n i i x x 1 1 (2 nd most common) 2 2-norm Eucledian norm Radius norm Distance norm = = n i i x x 1 2 2 i i x x x = (1 st most common and most useful norm) -norm Max-norm i n i x x 1 max = = (3 rd most common) Unit circles in various vector norms. 1.2. Matrix Norms The matrix norm is defined in terms of the effect of the matrix on a vector. If } { } ]{ [ y x A = When all combinations of { x } are considered, the combination that gives the largest ratio of x y is defined as the norm of matrix [ A ] = x Ax A max where {} } 0 { x for all combinations of { x }. p Name Formula 1 1-norm Cartesian norm = = = n i ij m j a A 1 1 1 max = maximum column sum
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2 -norm Max-norm = = = m j ij n i a A 1 1 max = maximum row sum 2 2-norm Spectral norm Radius norm For a square matrix: ( ) i n i A λ 1 2 max = = , where i is eigen value number i of the matrix. Eucledian norm Frobenius norm ∑∑ == = n i m j ij e a A 11 2 Note: e A A 2 1.3. Matrix Condition number 1 ]) ([ cond = A A A - For any matrix: 1 ]) ([ cond A - If = ]) ([ cond A , then [ A ] is singular. - A matrix with a large condition >>1 number is close to being singular. - A matrix with a condition number close to 1 is far from being singular. - For a diagonal matrix () ( ) i i d d A min / max ]) ([ cond = 1.4. Solving a system of Linear Algebraic equations Given a system of equations: } { } ]{ [ c x A = . We want to solve this system for the vector { x } and we know [ A ] and { c }. The following operations do not change the system of equations (i.e. the system of equation will have the same solution): - Any equation can be multiplied (or divided) by a non-zero scalar. - Any equation can be added to (or subtracted from) another equation. - The position of any 2 equations can be interchanged. 1.5. Gauss Elimination } { } ]{ [ c x A = If we can reduce the matrix [ A ] to an upper or lower triangular matrix, then we can easily solve for { x }. 1 1 1 1 , 1 3 13 2 12 1 11 c x a x a x a x a x a n n n n = + + + + + L 2 2 1 1 , 2 3 23 2 22 1 21 c x a x a x a x a x a n n n n = + + + + + L 3 3 1 1 , 3 3 33 2 32 1 31 c x a x a x a x a x a n n n n = + + + + + L 1 , 1 1 1 , 1 3 3 , 1 2 2 , 1 1 1 , 1 = + + + + + n n n n n n n n n n c x a x a x a x a x a L n n nn n n n n n n c x a x a x a x a x a = + + + + + 1 1 , 3 3 2 2 1 1 L
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3 = n n n n n n n n n n n n n n n n n n n n n n n n c c c c c x x x x x a a a a a a a a a a a a a a a a a a a a a a a a a 1 3 2 1 1 3 2 1 1 , 3 2 1 , 1 1 , 1 3 , 1 2 , 1 1 , 1 3 1 , 3 33 32 31 2 1 , 2 23 22 21 1 1 , 1 13 12 11 M M L L M M O M M M L L L Triangularization: 1- Subtract the first equation from the rest of the equations: () ( ) 11 21 1 1 1 1 , 1 3 13 2 12 1 11 2 2 1 1 , 2 3 23 2 22 1 21 a a c x
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NumericalMethodsChaper02 - 1. Numerical Methods - Chapter 2...

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