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Conjugate Gradient Methods

Conjugate gradient methods are tools for solving equations of the form "Ax = b." The variables "x" and "b" represent vectors, strings of numbers describing information, for example, the numbers showing the direction and force of a wind gust. "A" is a matrix, a table of numerical data. If the vectors or the matrix contain many numbers, conjugate gradient calculations become complex and lengthy, but computers handle the algorithms well.

Matrices
- A matrix consists of rows and columns of mathematical data. If you run a company with, for example, five stores, a matrix could show the sales in each store for each month of the year. What makes it different from a regular financial report is that matrices are set up for mathematical operations. You could, hypothetically, use a matrix to subtract last year's monthly sales from the matching squares in the current matrix to measure how much they've changed.
Steepest Descent
- If you wanted to determine "x" in "Ax=b", you could face a huge list of solutions, depending on how many figures you can plug into "A" and "b." Mathematics graphs the range of solutions as a bowl-shaped plane in space, where each point represents one solution to the equation; "x" represents the lowest point on the gradient of the curving plane. "Steepest Descent" refers to conjugate gradient methods for calculating that lowest point. This doesn't work for all forms of the equation, however.
Nonlinear
- Computer scientists employ nonlinear conjugate gradient methods in a number of disciplines, including engineering design and neural-net training. Using conjugate gradients on nonlinear equations becomes complicated fast: Some equations have multiple lowest points on the plane, and others don't actually have a lowest point. When using a computer to calculate the answers, some nonlinear methods require you to stop before getting an exact result: If you're too precise, the computation becomes too slow to be useful.
Conjugation
- Conjugate gradients get their name, in part, because the algorithms used to calculate them -- whether by hand or on a computer -- work as a series of approximations. First you make an approximate calculation of the gradient, then you make a conjugated, or related conjugation using the results of the first calculation. Finding "x" requires running the algorithms for solving the equation multiple times, getting closer each time. This multiple iteration of the equations makes conjugate gradient methods a natural for computers.

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