Function Repository Resource:

TridiagonalInverse

Source Notebook

Evaluate the inverse of a tridiagonal matrix

Contributed by: Jan Mangaldan

ResourceFunction["TridiagonalInverse"][a,b,c]

gives the inverse of the tridiagonal matrix with subdiagonal a, diagonal b and superdiagonal c.

Examples

Basic Examples (2)

The inverse of a 3×3 tridiagonal matrix:

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Multiply with the original tridiagonal matrix:

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$% . \!$\* TagBox[ RowBox[{"(", "", GridBox[{ { SubscriptBox["b", "1"], SubscriptBox["c", "1"], "0"}, { SubscriptBox["a", "1"], SubscriptBox["b", "2"], SubscriptBox["c", "2"]}, {"0", SubscriptBox["a", "2"], SubscriptBox["b", "3"]} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[ 0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "", ")"}], Function[BoxForm`e$, MatrixForm[ SparseArray[ Automatic, {3, 3}, 0, {1, {{0, 2, 5, 7}, {{2}, {1}, {1}, {2}, {3}, {3}, { 2}}}, {Subscript[$CellContext`c, 1], Subscript[$CellContext`b, 1], Subscript[$CellContext`a, 1], Subscript[$CellContext`b, 2], Subscript[$CellContext`c, 2], Subscript[$CellContext`b, 3], Subscript[$CellContext`a, 2]}}]]]]$ // Simplify // MatrixForm$

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Scope (1)

TridiagonalInverse can be used with numerical entries:

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Applications (2)

Construct a natural cubic spline interpolant to data:

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ya = {1., 2., 5., 7., 8., 5.};
n = Length[ya]; h = 0.5;
df = Differences[ya];
yp = ResourceFunction["TridiagonalInverse"][ConstantArray[1, n - 1], ArrayPad[ConstantArray[4, n - 2], 1, 2], ConstantArray[1, n - 1]] . (3/h Join[{df[[1]]}, ListConvolve[{1, 1}, df], {df[[-1]]}]);
iFun = Interpolation[Transpose[{List /@ (h Range[0, n - 1]), ya, yp}]]

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Plot the interpolant along with the data:

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Approximately solve the boundary value problem u_{{XMLElement[span, {class -> stylebox}, {x, XMLElement[i, {class -> ti}, {x}]}]}}+u=f;u(0)=u(1)=0:

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$f[x_] := Exp[-100 (x - 1/2)^2]; n = 100; h = 1./n; grid = N[Range[n - 1]]/n; ux = ResourceFunction["TridiagonalInverse"][ ConstantArray[1/h^2, n - 2], ConstantArray[1. - 2/h^2, n - 1], ConstantArray[1/h^2, n - 2]] . (f[grid]); ListPlot[Transpose[{grid, ux}]]$

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Show the error compared with the exact solution:

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Properties and Relations (1)

TridiagonalInverse can be faster than using Inverse on a tridiagonal matrix constructed with SparseArray and Band:

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$l1 = Table[ First[AbsoluteTiming[ With[{a = Array[\[Alpha], n - 1], b = Array[\[Beta], n], c = Array[\[Gamma], n - 1]}, Inverse[ SparseArray[{Band[{2, 1}] -> a, Band[{1, 1}] -> b, Band[{1, 2}] -> c}]]];]], {n, 3, 12}]; l2 = Table[First[ AbsoluteTiming[ With[{a = Array[\[Alpha], n - 1], b = Array[\[Beta], n], c = Array[\[Gamma], n - 1]}, ResourceFunction["TridiagonalInverse"][a, b, c]];]], {n, 3, 12}];$

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Version History

1.0.0 – 21 June 2021

Source Metadata

Citation:
- Usmani R.A., "Inversion of Jacobi’s tridiagonal matrix." Computers & Mathematics with Applications, vol. 27, no. 8, 59-66, 1994. DOI: 10.1016/0898-1221(94)90066-3
- Usmani R.A., "Inversion of a Tridiagonal Jacobi Matrix." Linear Algebra and its Applications, vol. 212–213, 413-414, 1994. DOI: 10.1016/0024-3795(94)90414-6

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License