MthSc 860
Spring 2007
Review for Exam 1


I. Scientific Computing
	Approximations
		Truncation error
		Roundoff error	
			chopping vs. rounding
		Absolute, relative, & percentage error	
		Condition number of f(x)	
		Forward vs. backward error	
		Stability of algorithm
	
	Computer Arithmetic
		Normalized floating point number system
		Representation in base 2
		Underflow, Overflow, Machine precision
		Arithmetic operations on floating point numbers
		Subtractive Cancellation

II. Systems of Linear Equations
		Fundamental Theorem of Linear Algebra
                Vector and Matrix Norms
		Condition number and relative error
		Condition number - properties
                Solving Linear Systems
			(naive) Gaussian Elimination
			Elementary elimination matrices
			LU factorization
			Pivoting
			Operation Count
			Direct methods for special systems
				Cholesky
				Banded algorithm
		
III. Linear Least Squares
	Least squares sol'n - def'n
        Application: curve fitting
	Background - concepts from Linear Algebra
		range, rank, null space
		orthogonality
		orthogonal invariance
	Least squares sol'n - methods
		Normal equations
		Sensitivity & Conditioning
		Orthogonalization methods
			QR
				Householder 
				Givens
			SVD
				numerical rank
				pseudoinverse