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Upgrading from:

LinearAlgebra`Tridiagonal`

Mathematica's built-in SparseArray function should be used to create tridiagonal matrices.
Mathematica kernel's built-in sparse solver LinearSolve has replaced TridiagonalSolve as a faster solver of tridiagonal matrices.

Suggestion for replacing Tridiagonal

This loads the package in Version 5.2:

Version 5.2 << LinearAlgebra`Tridiagonal`;

This defines the list that will give the nonzero diagonals of a matrix:

Version 5.2 {a, b, c} = {{7, 1, 11}, {4, 8, 2, 12}, {5, 9, 3}};
Wolfram Language code: {a, b, c} = {{7, 1, 11}, {4, 8, 2, 12}, {5, 9, 3}};

Here is the matrix m constructed from the list:

Version 5.2 m = Table[ Switch[j - i, -1, a[[j]], 0, b[[j]], 1, c[[j - 1]], _, 0], {i, 4}, {j, 4}];
Wolfram Language code: m = SparseArray[{Band[{1, 2}] -> c, Band[{1, 1}] -> b, Band[{2, 1}] -> a}, 4]

This displays the matrix using MatrixForm:

Version 5.2 MatrixForm[m];
Wolfram Language code: MatrixForm[Normal[m]]

Mathematica kernel's built-in sparse solver LinearSolve should be used in place of TridiagonalSolve:

Version 5.2 TridiagonalSolve[a, b, c, {2, 3, 4, 5}];
Wolfram Language code: LinearSolve[m, {2, 3, 4, 5}]