today’s class briefly summarize the first two parts error analysis roots of equation linear...

Today’s class

• Briefly summarize the first two parts• Error analysis• Roots of equation

• Linear Algebraic Equations• Gauss Elimination

Numerical Methods, Lecture 7 1

Prof. Jinbo Bi CSE, UConn

• Round-off errors are caused because exact numbers cannot be expressed in a fixed number of digits as with computer representations

• Round-off errors occurs from imprecision in representation of data

• Truncation errors result from a numerical approximation in place of an exact analytical formula• Finite divided difference, Infinite series

Error Analysis



• Bisection• 2 initial guesses, slow convergence, will not

diverge

• False-position• 2 initial guesses, slow-medium convergence,

will not diverge

Roots of equations



• Newton-Raphson• 1 initial guess, fast convergence, may

diverge

• Secant• 2 initial guesses, medium-fast convergence,

may diverge

Roots of equations



• Muller• 3 initial guesses, medium-fast convergence,

may diverge, only polynomials, complex roots

• Bairstow• 2 initial guesses, medium-fast convergence,

may diverge, only polynomials, complex roots

Roots of equations



Linear Algebraic Equations• Solving for roots gave us solutions to

equations of the form:

• A more general problem would be to solve the following n equations simultaneously

6Numerical Methods, Lecture 7 6


Linear Algebraic Equations• A linear algebraic system is a system of

equations where all the functions are linear



Linear Algebraic Equations• Graphical solutions

• Plot the functions and the solution is the intersection point of the functions

• For two dimensional linear systems, each equation is a line

• For three dimensional linear systems each equation is a plane



Linear Algebraic Equations• Example:



Linear Algebraic Equations• Singular system (no solution)



Linear Algebraic Equations• Singular system (infinite solutions)



Linear Algebraic Equations• Ill-conditioned system



Linear Algebraic Equations• Graphical methods work only for second

and maybe third order systems• Not precise• Useful visualization tool



Linear Algebraic Equations• In matrix form

• where A is a n x n matrix, and X and B are n x 1 vectors.



Matrices

• Definitions:• Symmetric matrix• Diagonal matrix• Identity matrix (I)



Matrices• Definitions:

• Upper triangular

• Lower triangular

• Banded

• Transpose



Matrix Operations

• Addition

• Subtraction

• Multiplication



• Addition/Subtraction - O(n2)• Multiplication - O(n3)

Matrix operations



• If A is non-singular and square, then A-1 is the inverse such that

Inverse Matrices



• In matrix form

• where A is a n x n matrix, and X and B are n x 1 vectors.

Linear algebraic equations



• We need to solve for X

Linear algebraic equations



Linear Equations

• How do we get A-1?• It is non-trivial• Not very efficient if solved by hand

• Usually use other methods to solve for X• Gauss Elimination• LU Decomposition



• Given a second-order matrix A, the determinant D is defined as follows:

• Given a third-order matrix A, the determinant D is defined as follows:

Determinants, Cramer’s Rule



• Using determinants to solve a linear system

• Cramer’s rule• Replace a column of coefficients in matrix A

with the B vector and find determinant

Determinants, Cramer’s Rule



25

Cramer’s rule example



• Extension of elimination of unknowns as a systematic algorithm

• Two steps• Elimination of unknowns• Back substitution

Gauss Elimination



• Forward elimination

• Eliminate x1 from row 2• Multiply row 1 by a21/a11

Gauss Elimination



• Eliminate x1 from row 2• Subtract row 1 from row 2

• Eliminate x1 from all other rows in the same way

• Then eliminate x2 from rows 3 to n and so on

Gauss Elimination



• Forward elimination

• Back substitute to solve for x

Gauss Elimination



• Back substitution

• In general,

Gauss Elimination



Gauss Elimination



• Computational complexity• 2n3/3 + O(n2)• three orders of increase for every order of

increase in n• Most of the effort is incurred in the

elimination step

Gauss elimination



• Things to worry about• Division by zero• Round-off error• Ill-conditioned system

Gauss elimination



• Ill-conditioned system example

Gauss elimination



• If the determinant is close to zero, the system is ill-conditioned

• If the determinant is exactly zero, the system is singular

• It is difficult to specify how close to zero, as the magnitude of the determinant can be changed by multiplying by a constant without changing the solution

Gauss elimination



• Basic idea is to remove divide by zero if a11 is zero

• Swap the row with the largest element with the top row

Gauss elimination with pivoting



• It is sometimes useful to scale the equations so that the largest coefficient in any row is 1

• Example

Gauss elimination with scaling



• Example

Gauss elimination with scaling



Next class

• LU Decomposition• Read Chapter 10



today’s class briefly summarize the first two parts error analysis roots of equation linear...

Documents

jinbo bi cse

solutionnumerical methods

planenumerical methods

linearnumerical methods

mediumfast convergence

divergesecant2 initial

uconnmuller3 initial

slow convergence