Tri-Diagonal Linear Systems

Theorem  Suppose that the tri-diagonal linear system    has    and   for .  If   ,  and    ,  and  , then there exists a unique solution.  

    The enumeration scheme for the elements in (1) do not take advantage of the overwhelming number of zeros.  If the two subscripts were retained the computer would reserve storage for   elements.  Since practical applications involve large values of  , it is important to devise a more efficient way to label only those elements that will be used and conceptualize the existence of all those off diagonal zeros.  The idea that is often used is to call elements of the main diagonal , subdiagonal elements   are directly below the main diagonal, and the superdiagonal elements    are directly above the main diagonal.  Taking advantage of the single subscripts on we can reduce the storage requirement to merely   elements.  This bizarre way of looking at a tridiagonal matrix is  

    .

Observe that A does not have the usual alphabetical connection with its new elements named  , and that the length of the subdiagonal and superdiagonal is n-1.

 

Definition (tri-diagonal system).  If A is tridiagonal, then a tridiagonal system is

(2)    

A tri-diagonal linear system    can be written in the form    

(3)    

								=
								=
								=
								=
								=

     

Goals

To understand the "Gaussian elimination process."  
To understand solutions to"large matrices" that are contain mostly zero elements, called "sparse matrices."  
To be aware of the uses for advanced topics in numerical analysis:  
(i)    Used in the construction of cubic splines,
(ii)   Used in the Finite difference solution of boundary value problems,  
(iii)  Used in the solution of parabolic P.D.E.'s.

 

Algorithm (tridiagonal linear system).  Solve a tri-diagonal system    given in (2-3).  
Below diagonal elements:    

  
Main diagonal elements:       
Above diagonal elements:       
Column vector elements:         

Mathematica Subroutine