tridiagonal

tridiagonal is a C++ template library with solvers for tridiagonal block matrix systems.

Solvers

tridiagonal::solve_tridiagonal solves block matrix systems on the form

tridiagonal::solve_off_tridiagonal solves block matrix systems on the form

Installation

Eigen is used for linear algebra structures and operations.

To install the library clone the repository and copy the tridiagonal directory to the desired install location.

Example

//
//        LHS       X   =  RHS
//
//  | b0  c0  0  | |x1|   |d0|
//  | a1  b1  c1 | |x2| = |d1|
//  | 0   a2  b2 | |x3|   |d2|
//

#include <eigen3/Eigen/Dense>
#include <tridiagonal/tridiagonal>

int main(){

    Eigen::Matrix<double, 6, 6> LHS;

    LHS <<    0, 3,  1, 2,  0, 0,
              2, 1,  4, 5,  0, 0,

              6, 7,  1, 2,  4, 5,
              1, 3,  9, 0,  2, 3,

              0, 0,  2, 1,  5, 2,
              0, 0,  3, 7,  9, 3;

    Eigen::Matrix<double, 6, 1> RHS;
    RHS <<    0,
              2,
              3,
              1,
              -9,
              2;

    Eigen::Matrix2d a1, a2, b0, b1, b2, c0, c1;
    b0 = LHS.block(0, 0, 2, 2);
    b1 = LHS.block(2, 2, 2, 2);
    b2 = LHS.block(4, 4, 2, 2);

    c0 = LHS.block(0, 2, 2, 2);
    c1 = LHS.block(2, 4, 2, 2);

    a1 = LHS.block(2, 0, 2, 2);
    a2 = LHS.block(4, 2, 2, 2);

    Eigen::Vector2d d0, d1, d2;
    d0 = RHS.block(0, 0, 2, 1);
    d1 = RHS.block(2, 0, 2, 1);
    d2 = RHS.block(4, 0, 2, 1);

    vector<Eigen::Matrix2d> diagonal = {b0, b1, b2};
    vector<Eigen::Matrix2d> lower_diagonal = {a1, a2};
    vector<Eigen::Matrix2d> upper_diagonal = {c0, c1};
    vector<Eigen::Vector2d> rhs = {d0, d1, d2};

    vector<Eigen::Vector2d> solution = tridiagonal::solve_tridiagonal(lower_diagonal,
                                                                      diagonal,
                                                                      upper_diagonal,
                                                                      rhs);

    Eigen::Matrix<double, 6, 1> tridiagonal_X;
    tridiagonal_X.block(0, 0, 2, 1) = solution[0];
    tridiagonal_X.block(2, 0, 2, 1) = solution[1];
    tridiagonal_X.block(4, 0, 2, 1) = solution[2];

    Eigen::Matrix<double, 6, 1> eigen_X = LHS.fullPivHouseholderQr().solve(RHS);

    assert(tridiagonal_X.isApprox(eigen_X));
}

Tests

The solvers are unit tested with Catch2. To compile and run the unit tests make sure that tridiagonal, Eigen and Catch2 are in the include path and run

$ gcc tests/*.cpp -o test.out -std=c++11
$ ./test.out