Matrix Solution to Linear Equations and Maikov Chains

DIRECT SOLUTION AND CONVERGENCE METHOD

Before computer programs offered readj solutions, problems such as Markov chains were solved in a direct manner, by algebraically manipulating the equations. This direct solution requires an understanding of simple mafrix arithmetic, and very careful attention to calculating the numbers correctly. The convergence method is now easier although it requires many more calculations. Without the computer, we would never even consider using this approach-with a computer, it is the best choice.

GENERAL MATRIX FORM

A mafrix is a rectangular arrangement of elements into rows and columns A matrix A is said to be m x n (pronounced "m by n") if there are m rows and n columns in A

Certain properties of a maUix make it a valuable tool for solving simultaneous linear equations. These elementarj mafrix operations, called transformations, allow you to alter the rows (which will represent equations) without changing the solution. There are three basic row operations:

1. Multiplication or division of all elements of the row by any number.

2. Interdianging of any two rows (and consequently of all rows)

3. The addition or si4)traction of the elements of one row with the corresponding elements of another.

To relate the maUix to simultaneous linear equations, consider a three-equation example, where only the coefficients, a, remain as unknowns:

The ihm. tkiucnijry n ii. ur

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tDV¹«hcr Pniblbilitie F.rxiiM « ,

. -ill tqujj (I . l)di 1 1 . tbt cgmiiaiu

(Hii!)

Computer Program Direct Solution

The following FORTRAN program accepts a sjstem of 10 equations in 10 variables (unknowns) and applies the maUix method of solution. This is done using the method of Gaussian elimination, discussed in the previous section.