Where are linear equations with large number of variables used?

Question

Do weather prediction / financial models or missile / rocket trajectory prediction model use these equations? What method or algorithm is used for the same?

A long time ago, I read that constructing a wing of a plane requires a linear equation system of 100,000 unknowns, if I remember right. Weather prediction is probably too complex to deal with it using linear equations, but the other applications might work. — Peter, Jul 06 '14 at 09:06
For very large equation systems, the gauss-elimination-method is not very useful, in this case numerical methods are used.In Practice, matrices often have some structure or many zeros, making the solution easier. — Peter, Jul 06 '14 at 09:38

score 2 · Answer 1 · answered Jul 07 '14 at 05:45

Where would linear equations not be used? For example in factorization of large integers? Well, there is Fermat's method: If yho have $x,y$ with $x^2\equiv y^2\pmod N$, then $N\mid (x+y)(x-y)$ so $\gcd(N,x\pm y)$ may happen to be proper divisors of $N$. So far, so good, but subtraczing (multiples of) $N$ from some $x^2$ will hardly ever produce a square. But if you have found enough near-squares, i.e. $x_i^2-k_iN=y_i^2z_i$ where the $z_i$ contain only "some" small primes, then you may hope to combine these in a way to produce a perfect square. That is: We try to find a nontrivial element in the kernel of a matrix over $\mathbb F_2$ of size number-of-primes-called-small.

score 0 · Answer 2 · answered Jul 07 '14 at 04:55

The first computer program to describe a linear model of the economy was written at Harvard by Wassily Leontief. It was written with 500 equations in 500 unknowns. That was too many for the computer so he broke it down into 42 equations with 42 unknowns, I don't know what algorithm he used to solve this but its probably worth looking it up.

score 0 · Answer 3 · answered Jul 07 '14 at 05:21

Nonlinear systems of equations are solved using Newton's method. Guess what - at its core linear equations have to be sold.

Any computation to solve problems with huge number of unknowns (weather prediction, earthquake inversions, any simulation of physical systems) has to solve linear equations.

These equations are solved by iterative methods that do not require access to the complete matrix, but only need matrix times vector multiplications.

Where are linear equations with large number of variables used?

3 Answers3