Answer to Question #204412 in Calculus for smi

Question

What is the resemblance between linear BVPs and systems of algebraic equations Ax=b?

Answer 1

For simplicity of exposition, let’s look at the case where p(x) is a constant, say p(x) = 1, and q = 0. Then we have the simplified difference equation 

$\dfrac{−U_{i+1} + 2U_i − U_{i−1}}{h^2}= f(x_i)\space\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space i = 1, . . . , n$

at each interior grid point x1, x2, . . . , xn. Multiplying by h² produces

$−U_{i−1} + 2U_i − U_{i+1} = h^2 f(x_i)\space\space\space\space\space\space\space\space\space\space\space\space\space\space i = 1, . . . , n$

The corresponding matrix problem is AU = f where A is the matrix.

and $U =(U_1,U_2,U_3.........U_n)^T, f=h^2(f(x_1),f(x_2),........,f(x_n))^T$  

Dirichlet boundary data. Clearly this matrix is symmetric and tridiagonal; in addition, it can be shown to be positive definite so the Cholesky factorization A = LL^T for a tridiagonal matrix can be used. Recall that tridiagonal systems require only O(n) operations to solve and only three vectors must be stored to specify the matrix. In our case the matrix is symmetric and so only two vectors are required; this should be contrasted with a full n × n matrix which requires n² storage and O(n³ ) operations to solve.