The Conjugate Gradient Method
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
October 21, 2010
The Conjugate gradient method
Restricted to symmetric positive denite n n systems
Ax = b.
Start with x(0) Rn . G
The Singular Value Decomposition
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
October 4, 2010
Applications of SVD
solving over-determined equations
statistics, principal component analysis
numerical deter
Lecture4 INF-MAT 4350 2010: 5. Fast Direct
Solution of Large Linear Systems
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
September 16, 2010
Test Matrix T1 Rm,m
d
a
0
a d
a
. . .
.
.
.
T1 := 0
0
a d a
0 a
Computing Eigenvalues and/or
Eigenvectors;Part 1, Generalities and
symmetric matrices
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
November 10, 2010
Today
Given a matrix A Cn,n .
Finding the eigenvalues u
The Classical Iterative Methods
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
October 14, 2010
Jacobis Method (J method)
Solve Ax = b, A Cn,n , b Cn .
Assume A has nonzero diagonal elements. Solve for xi .
Lecture 2 INF-MAT 4350 2010: 3.1-3.7, LU,
symmetric LU, PLU, Positve (semi)Denite,
Cholesky, Semi-Cholesky
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
September 2, 2010
LU of Tridagonal matrix
d1
a2
c1
d
Orthonormal Transformations
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
October 25, 2010
Applications of transformation Q : Rm Rm ,
with QT Q = I
1.
2.
3.
4.
solving
solving
solving
nding
linear equation
Vector and Matrix Norms
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
October 4, 2010
Vector Norms
Denition (Norm)
A norm in Rn (Cn ) is a function : Rn (Cn ) R that
satises for all x, y in Rn (Cn ) and al
Lecture 3 INF-MAT 4350 2010: 4: Test
Problems and Kronecker products
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
September 3, 2010
Numerical solution of the Poisson Problem
u=0
(u +u ) =f
xx yy
u=0
u=0
u
Lecture 5 INF-MAT 4350 2010: Eigenpairs
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
September 24, 2010
Eigenpair
Suppose A Cn,n is a square matrix, C, x Cn \ cfw_0,
and Ax = x.
is an eigenvalue, x is an
Least Squares
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
October 26, 2010
Linear system
Linear system Ax = b, A Cm,n , b Cm , x Cn .
under-determined (m < n): No solution, or an innite
number of solutio
Lecture 1 INF-MAT 4350 2010: Chapter 2.
Examples of Linear Systems
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
August 26, 2010
Notation
The set of natural numbers, integers, rational numbers, real
number
gram3mi_s.1wkm._4;ug_yw%rnms.
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ML_ h _. ._
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W m. ._._.
' u MM +
, 32" "WW . 8+3x3y7) -ln(x3+3x2y~2xy2) ln(x23xy-4y ) + e
(5-2)? +5. grad F = 25uv(u +1), where E = xi+yi,
x6 +3x2y4 Sy6 ,
3x2y
6 6 [
u= x+3
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Computing Eigenvalues and/or
Eigenvectors;Part 2, The Power method
and QR-algorithm
Tom Lyche
Centre of Mathematics for Applications,
Department of Informatics,
University of Oslo
November 19, 2010
Today
The
The
The
The
The
power method to nd the dominant