Tridiagonalmatrix

In linear algebra is a tridiagonal matrix ( three ribbon matrix ) is a square matrix that contains entries only in the diagonals and secondary diagonals in the first two non-zero. Tridiagonal matrices occur in the numerics on quite often, for example in the calculation of cubic splines in the discretization of the second derivative to one-dimensional areas (especially for Sturm-Liouville problems ), in the computation of orthogonal polynomials and functional systems (such as in the calculation of Bessel functions ) and Krylov subspace method based on three- term recursion.

Definition

A matrix is called tridiagonal if it has the following form:

It applies to everyone. A tridiagonal matrix is called unreduced or irreducible if the elements are equal to zero in the secondary diagonals all, that is all true. Are the main and off-diagonal entries constant, that is true, and so one speaks of a tridiagonal Toeplitz matrix.

Properties

A tridiagonal matrix is both a special case of a band matrix and a Hessenberg matrix. A diagonally dominant tri-diagonal matrix is always regular.

Systems of linear equations with a tridiagonal matrix can be solved efficiently with a cost of O (n). Either with the very fast Thomas algorithm or stability problems with the help of the Gaussian process with Pivoting. Systems of equations with tridiagonal matrices can be calculated with relatively large dimension by means of a direct solver so themselves.