Here are ten things about numerical linear algebra that you may find surprising if you’re not familiar with the field.
- Numerical linear algebra applies very advanced mathematics to solve problems that can be stated with high school mathematics.
- Practical applications often require solving enormous systems of equations, millions or even billions of variables.
- The heart of Google is an enormous linear algebra problem. PageRank is essentially an eigenvalue problem.
- The efficiency of solving very large systems of equations has benefited at least as much from advances in algorithms as from Moore’s law.
- Many practical problems — optimization, differential equations, signal processing, etc. — boil down to solving linear systems, even when the original problems are non-linear. Finite element software, for example, spends nearly all its time solving linear equations.
- A system of a million equations can sometimes be solved on an ordinary PC in under a millisecond, depending on the structure of the equations.
- Iterative methods, methods that in theory require an infinite number of steps to solve a problem, are often faster and more accurate than direct methods, methods that in theory produce an exact answer in a finite number of steps.
- There are many theorems bounding the error in solutions produced on real computers. That is, the theorems don’t just bound the error from hypothetical calculations carried out in exact arithmetic but bound the error from arithmetic as carried out in floating point arithmetic on computer hardware.
- It is hardly ever necessary to compute the inverse of a matrix.
- There is remarkably mature software for numerical linear algebra. Brilliant people have worked on this software for many years.
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Sean Devlin 01.20.10 at 10:53
No love for ATLAS? I thought this was use in favor of lapack nowadays.
http://math-atlas.sourceforge.net/
Giles Warrack 01.20.10 at 11:31
Re No 3, Herbert Wilf (Penn) has a nice handout on the Kendall-Wei algorithm and Google
http://www.math.upenn.edu/~wilf/
joeyo 01.20.10 at 12:27
@Sean: LAPACK is built on top of BLAS/ATLAS.
Evan 01.20.10 at 22:48
So.. you have convinced me to learn Linear Algebra. Do you have any suggestions for books to get started? Thanks for the inspiration!
John 01.21.10 at 00:11
Evan, there are a lot of linear algebra books out there. I got a review copy of Cheney and Kincaid’s linear algebra book a few weeks ago and I thought it was a nice text. It’s easy to read and has a good mix of theory and application.
If you know basic linear algebra and want to jump into numerical linear algebra, I recommend Demmel’s Applied Numerical Linear Algebra. Also, there’s the classic Matrix Calculations by Golub and Van Loan.
Evan 01.21.10 at 14:54
John, thanks for your quick reply. I’ll definitely check those out!
Matt 01.24.10 at 06:46
By the way, has anyone here played with GotoBLAS?
Is its performance really as good as the benchmarks on its website suggest or is it just advertisement?
The source code is freely available, though only for research applications (commercial is not free). Here are relevant links:
http://www.tacc.utexas.edu/resources/software/gotoblasfaq/
http://en.wikipedia.org/wiki/Kazushige_Goto
http://www.tacc.utexas.edu/tacc-projects/
http://web.tacc.utexas.edu/~kgoto/
http://www.pathscale.com/building_code/gotoblas.html
Michael Holmes 04.01.10 at 08:33
John suggested I post this here –
We recently released a MATLAB package called QLA that dramatically accelerates core linear algebra routines on dense matrices. It includes SVD, linear systems, least squares, PCA, etc., and speedups range from 10x to 1,000x+ on benchmarks. The gains come by trading a minuscule bit of accuracy for a mammoth increase in speed.
You can find more details and the free beta version here:
http://massiveanalytics.com.
Ron Modesitt 07.26.10 at 18:46
What a thoroughly interesting blog! Glad I found you, with thanks to Stumble.
Ron
Luis 07.19.11 at 17:45
Another interesting surprise is that, often, the matrices involved are very sparse.
Books: I love “Matrix algebra useful for statistics” by Searle.
Utkarsh Upadhyay 09.13.11 at 11:02
Some surprises indeed!
However, a small note for point 4:
At least solutions for spare system of equations have benefited two orders of magnitude more from algorithmic developments than hardware advancements (slide 7) to a total of 10 orders of magnitude improvement from 1970 to 2001.
For other kinds of linear equations, the improvements indeed have been approximately the same..
fish 09.13.11 at 11:32
Ha, good to know re: matrix inversion — I often use inverts in calculating color transforms (see http://www.brucelindbloom.com/index.html?Eqn_RGB_XYZ_Matrix.html) but hey. Thanks!
Juan Solano 09.13.11 at 11:33
Let me recommend the videos in Khan Academy, is a very nice place to start and refresh the key concepts.. and then digest the Linear Algebra beauty…
http://www.khanacademy.org/#browse
Juan
J 09.13.11 at 12:18
Charles Van Loan has a sweet pair of custom Nike kicks. Not even joking.
martin 09.13.11 at 15:53
Could you elaborate on number 6? I would have thought it would have taken close to a millisecond to just read a million values.
mg 09.13.11 at 17:07
@Evan my favorite text is Janich’s Linear Algebra.
Evan 09.13.11 at 17:16
Thanks mg. I’ll check that out as well.
AV 09.14.11 at 04:20
this article is a ‘trivality’.
any indian schoolkid knows this.
– of course linear algebra is balzing fast for FEA and related computations.
but the writer should state that the ’speed’ in fast lin algebra comes not from inherent properties of lin-algebra, but from the fact that we MASSIVELY approximate the actual behavior of the beams/trusses/etc. we approximate reality. and thats becoz we dont have good computational algos to handle the ‘true’ mathematical representations.
if that was ‘overhead transmission’ – then here’s a simplification– try to find a ‘fast’ solution to the Navier Stokes equations using ‘linear algebra’ – in their pure form.
Anselmo 11.19.11 at 20:48
For me, the greatest achievement of linear algebra is the simplex method. Linear optimisation problems with million variables may be solved in home computers in a few minutes. It is remarkably efficient, though it has an exponential worst case performance. In addition, most exact non-linear optimisation algorithms use the simplex method for solving the linear approximation model.
babita 05.19.12 at 09:02
i want to solve the equation Ax=B,where A is sparse given in CSC format.A and B are known.
Can you give me some references to solve it???
Reply as soon as possible…its urgent..