Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [numerical-linear-algebra]

Questions on the various algorithms used in linear algebra computations (matrix computations).

Questions tagged with this tag can be about, but not limited to:

  1. Matrix decompositions like SVD, QR, Cholesky, etc.
  2. The solution of linear systems and least squares problems.
  3. Analysis of numerical linear algebra algorithms like condition numbers and stability analysis.
  4. Eigenvalue problems.
  5. The designs of direct or iterative methods to solve linear systems.
3541 questions
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Non-monotonic decrease of residuals in Conjugate Gradients:

In some of my numerical programming using conjugate gradient solvers, I noticed an alarming problem: The residuals were not monotonically decreasing to zero, but were sometimes increasing. In this document, the author writes About fifteen years ago…
user14717
  • 4,902
8
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1 answer

What is the operation count for QR factorization using Householder transformations?

I have a hard time finding the operation count of QR factorization when using Householder transformations. The answer is $2mn^2 - \frac{2n^3}{3}$, but have no clue on how to get this count following some procedure. Could anyone help me go through…
onimoni
  • 6,376
6
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IEEE 754 as a mathematical space

Integer operations in computers (i.e. 32-bit integers) probably can be represented best by modular arithmetic (because of integer overflows/underflows). What about IEEE 754 floating point arithmetic? You have rounding and negative zeros. What would…
user133801
  • 61
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1 answer

Finding generalized eigenvalues with linear constraints

I have a generalized eigenvalue problem $$Mx = \lambda Bx$$ with the additional constraint that $Cx=0$, where $M$ and $B$ are positive-definite and $C$ is a sparse and rectangular. Is there a simple way of solving for the generalized spectrum…
user7530
  • 49,280
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2 answers

Exact Computational Costs/Flop count for algorithms

I need exact computational costs for different algorithms to benchmark a code. For instance, the exact cost of Gauss Elimination is given here. I am not interested just in the leading order term. The algorithms for which I need the exact cost…
user17762
5
votes
1 answer

Proving a property about Gauss-Seidel

This is a homework problem, so please give hints or tips instead of full answers. The problem is as follows: Let $G$ be the iteration matrix of the Gauss-Seidel method; i.e. $$G=I-(D-L)^{-1} A$$ where $D-L$ is the lower triangular part of $A$,…
ithisa
  • 2,763
4
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1 answer

How to recover Q from the (tiled) QR decomposition using householder factorisation?

I'm trying to implement the tiled QR decomposition in MATLAB (in an attempt to understand it), and I'm trying to check that my SGEQRF (upper corner tiles) function is working correctly. I have a similar C implementation that does work correctly (for…
LonelyCat
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Find the eigenvector with maximum overlap

Given a large symmetric matrix $A$, there are methods to find the largest or smaller eigenvalue, or the eigenvalue closest to some initial value. Is there any method to find the normalized eigenvector with maximum overlap (dot product) with an…
Jellby
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Effects of elementary row operation on condition number

How does any elementary row operation on a matrix affect the condition number? Can an ill conditioned matrix be improved by just some elementary row operations? Can I improve the accuracy of solving linear system by some row operations?
lakshmi
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$A, B$ sparse imply $AB$ is sparse?

Today I read the claim that if $A$ and $B$ are sparse matrices, then $AB$ is also sparse. I didn't believe it at first, but could not exhibit a counterexample. So is this claim in fact true? If so, how sparse is $AB$? Can a nice result like ``if $A$…
user14717
  • 4,902
3
votes
1 answer

Is this a reasonable method of numerically comparing two matrix functions?

I am currently trying to compare two matrices with elements which are too complicated for me to algebraically show that they are equal element wise and I decided to try the following approach: Suppose I have two matrix functions $A(t)$ and $B(t)$…
JessicaK
  • 7,655
3
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0 answers

What's the state of the art for computing the largest singular value of a matrix

My matrix is not sparse, and is sized 30k by 30k. Most importantly, the gap between the largest and the second largest singular values is small or even 0. ARPACK, SLEPc, Matlab, PROPACK? Which one is more stable and fast? Many thanks.
Jonathan
  • 31
3
votes
1 answer

How to Store a Banded Matrix by Diagonal

I'm taking a graduate level independent study course this semester in Matrix Computations. I'm not getting much support from the professor, so am turning to the excellent StackExchange community for help (how's that for buttering you up). The…
Dave Kincaid
  • 131
3
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2 answers

Householder Vector algorithm in Golub and Van Loan

In the 4th edition of "Matrix Computations", Golub and Van Loan present "Algorithm 5.1.1 (Householder Vector)". The first couple of lines (translated into MATLAB-syntax) read: m = length(x); sigma = x(2:m)'*x(2:m); v = [1; x(2:m)]; if sigma == 0 &&…
Jamie Ballingall
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Fast computation of eigenvectors of perturbed matrix

Given a real symmetric matrix $A$, say we already know its spectral decomposition $A = UDU^T$. We wish to compute the spectral decomposition of $A+E$, where $E$ is a small matrix (I am particularly interested in the case where E is nonzero in only…
Van Latimer
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