Lecture 9
Conditioning and Stability I
MIT 18.335J / 6.337J
Introduction to Numerical Methods
Per-Olof Persson ([email protected])
October 10, 2007
1
Conditioning
Absolute Condition Number of a differentiable problem f at x:
= sup
x
f
= J(x)
x
where the J
Notes on Vector and Matrix Operations
Robert A. van de Geijn
Department of Computer Science
The University of Texas at Austin
Austin, TX 78712
[email protected]
September 2, 2014
1
(Hermitian) Transposition
In the following, assume that C, x Cn , and A C
Notes on Gram-Schmidt QR Factorization
Robert A. van de Geijn
Department of Computer Science
The University of Texas
Austin, TX 78712
[email protected]
October 16, 2014
A classic problem in linear algebra is the computation of an orthonormal basis for th
Notes on Conditioning
Robert A. van de Geijn
The University of Texas
Austin, TX 78712
October 16, 2014
NOTE: I have not thoroughly proof-read these notes!
1
Motivation
Correctness in the presence of error (e.g., when oating point computations are performe
Notes on Conditioning
Robert A. van de Geijn
The University of Texas
Austin, TX 78712
October 6, 2014
NOTE: I have not thoroughly proof-read these notes!
1
Motivation
Correctness in the presence of error (e.g., when oating point computations are performed
Notes on Eigenvalues and Eigenvectors
Robert A. van de Geijn
Department of Computer Science
The University of Texas
Austin, TX 78712
[email protected]
October 31, 2014
If you have forgotten how to nd the eigenvalues and eigenvectors of 2 2 and 3 3 matric
Notes on Gram-Schmidt QR Factorization
Robert A. van de Geijn
Department of Computer Science
The University of Texas
Austin, TX 78712
[email protected]
September 15, 2014
A classic problem in linear algebra is the computation of an orthonormal basis for
Chapter 4
Singular Value
Decomposition
In order to solve linear systems with a general rectangular coecient matrix, we
introduce the singular value decomposition. It is one of the most important tools
in numerical linear algebra, because it contains a lot
Lecture 3 Part I: Norms
Robert A. van de Geijn
The University of Texas at Austin
Numerical Analysis: Linear Algebra Fall 2011
vertiujilogo
http:/z.cs.utexas.edu/wiki/rvdg.wiki/
1
Vector norms
http:/z.cs.utexas.edu/wiki/rvdg.wiki/
2
Denition of a vector no
NEED TO tMPose ZES'HZLC‘UON} 0N NoEM
a? X a
2
\ l _ ’
1.4.48?" 9 Wm ‘7: ll AX +2“ XUZ
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residual.
ar duh; msvncxlrlt
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Notes on Vector and Matrix Norms
Robert A. van de Geijn
Department of Computer Science
The University of Texas at Austin
Austin, TX 78712
[email protected]
September 15, 2014
1
Absolute Value
Recall that if C, then | equals its absolute value. In other w
Notes on Vector and Matrix Norms
Robert A. van de Geijn
Department of Computer Science
The University of Texas at Austin
Austin, TX 78712
[email protected]
September 11, 2014
1
Absolute Value
Recall that if C, then | equals its absolute value. In other w
Lecture 8 - Floating Point Arithmetic,
The IEEE Standard
MIT 18.335J / 6.337J
Introduction to Numerical Methods
Per-Olof Persson ([email protected])
October 3, 2007
1
Floating Point Formats
Scientic notation:
1.602 10 19
sign
signicand
base
exponent
Floa
Lecture 3
The Singular Value Decomposition
MIT 18.335J / 6.337J
Introduction to Numerical Methods
Per-Olof Persson ([email protected])
September 12, 2007
1
The SVD - The Main Idea
Motivation:
The image of the unit sphere under any m n matrix is a hyperelli
Lecture 4
The QR Factorization
MIT 18.335J / 6.337J
Introduction to Numerical Methods
Per-Olof Persson ([email protected])
September 17, 2007
1
Projectors
A projector is a square matrix P that satises
P2 = P
Not necessarily an orthogonal projector (more l
Lecture 5
Gram-Schmidt Orthogonalization
MIT 18.335J / 6.337J
Introduction to Numerical Methods
Per-Olof Persson ([email protected])
September 19, 2007
1
Gram-Schmidt Projections
The orthogonal vectors produced by Gram-Schmidt can be written in
terms of pr
Lecture 2
Orthogonal Vectors and Matrices, Norms
MIT 18.335J / 6.337J
Introduction to Numerical Methods
Per-Olof Persson ([email protected])
September 10, 2007
1
Transpose and Adjoint
For real A, the transpose of A is obtained by interchanging rows/columns
Lecture 1
Introduction, Basic Linear Algebra
MIT 18.335J / 6.337J
Introduction to Numerical Methods
Per-Olof Persson ([email protected])
September 5, 2007
1
Matrix-Vector Multiplication
Matrix-vector product b = Ax
n
bi =
aij xj ,
i = 1, . . . , m
j=1
Lin
Notes on the Singular Value Decomposition
Robert A. van de Geijn
The University of Texas at Austin
Austin, TX 78712
September 18, 2014
NOTE: I have not thoroughly proof-read these notes!
We recommend the reader review Weeks 9-11 of Linear Algebra: Foundat
Notes on the Singular Value Decomposition
Robert A. van de Geijn
The University of Texas at Austin
Austin, TX 78712
September 11, 2014
NOTE: I have not thoroughly proof-read these notes!
We recommend the reader review Weeks 9-11 of Linear Algebra: Foundat
Notes on Numerical Stability
Robert A. van de Geijn
The University of Texas
Austin, TX 78712
October 16, 2014
Based on
Goal-Oriented and Modular Stability Analysis [3, 4]
by
Paolo Bientinesi and Robert van de Geijn
NOTE: I have not thoroughly proof-read t
Notes on Numerical Stability
Robert A. van de Geijn
The University of Texas
Austin, TX 78712
October 10, 2014
Based on
Goal-Oriented and Modular Stability Analysis [3, 4]
by
Paolo Bientinesi and Robert van de Geijn
NOTE: I have not thoroughly proof-read t
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