Introduction to the algorithm note 28th Chapter matrix Operations

28.1 solving a linear equation group definition

Singular matrix: a matrix with rank not full rank

(linear equations): the rank of matrix vector A of a linear equation group is less than n

Over-set (linear equations): The number of equations exceeds the number of unknown variables n

In fact, I have forgotten almost. = Pick it up quickly

Lup Analysis

Find three nxn matrices L, U, and P to meet

Pa=lu

What's the relationship with program assignment?

where L (lower) is the lower triangular matrix, U (upper) is the upper triangular matrix and P is a permutation matrix. We call the Lup decomposition of the matrix Lup, which satisfies the upper formula, called a.

There is such decomposition for all non-singular matrices. In the following pseudo code the transpose matrix is represented by the array π.

Calculate an LU decomposition

Considering the non-singular nxn matrix A, the Gaussian elimination method is used to create a LU decomposition.

Use recursive return to dissolve

This is called a for A11 's shure (really nasty =. =

In Lup decomposition, the element we are removing is called the principal element, for the diagonal of the matrix U, in order to avoid the 0 as a divisor, we include a transpose matrix p in the Lup decomposition, so the operation is called the Select Principal element.

Here's the code that uses iterations instead of tail recursion:

Run time is cubic

Calculate a Lup decomposition

And Lu decomposition is similar

Introduction to the algorithm note 28th Chapter matrix Operations