# Linear Solvers

This section covers the solvers available in geometry central for sparse linear algebra problems.

All solvers support real and complex matrices, and can be templated on a float, double, or std::complex<double>.

#include "geometrycentral/numerical/linear_solvers.h"

## Direct solvers

These solvers provide a simple interface for solving sparse linear $Ax = b$.

A key feature is that these solvers abstract over the underlying numerical library. In their most basic form, Eigen’s sparse solvers will be used, and are always available. However, if present, the more-powerful Suitesprase solvers will be used intead. See the dependencies section for instruction to build with Suitesparse support.

As always, be sure to compile with optimizations (i.e. set CMAKE_BUILD_TYPE to Release) for meaningful performance. In particular, Eigen’s built-in solvers will be very slow in debug mode (though the Eigen QR solver is always slow).

### Quick solves

These are one-off routines for quick solves.

Vector<T> solve(SparseMatrix<T>& matrix, const Vector<T>& rhs)

Solve a system with a general matrix. Uses a QR decomposition interally.

Warning: The Eigen built-in sparse QR solver is very inefficient for many problems. Also, it doesn’t work well for underdetermined systems.

Vector<T> solveSquare(SparseMatrix<T>& matrix, const Vector<T>& rhs)

Solve a system with a square matrix. Uses an LU decomposition interally.

Vector<T> solvePositiveDefinite(SparseMatrix<T>& matrix, const Vector<T>& rhs)

Solve a system with a symmetric positive (semi-)definite matrix. Uses an LDLT decomposition interally.

### Retain factorizations

When solving many linear systems $Ax=b$ with the same matrix $A$ but different $b$, it is dramatically more efficient to retain and reuse the factorization of $A$. The following solver classes are stateful, storing the factorization to be re-used for may solves.

SparseMatrix<double> A = /* ... some matrix ... */;

// Build the solver
Solver<double> solver(A);

// Solve a problem
Vector<double> rhs1 = /* ... */;
Vector<double> sol = solver.solve(rhs1);

// Solve another problem
Vector<double> rhs2 = /* ... */;
Vector<double> sol2 = solver.solve(rhs2);

// Can place solution in existing vector
Vector<double> rhs3 = /* ... */;
solver.solve(sol, rhs3);

// Some solvers have extra powers.
// Solver<> can compute matrix rank, since it uses QR under the hood.
std::cout << "matrix rank is " << solver.rank() << std::endl;

template <typename<T>> class Solver

Solve a system with a general matrix. Uses a QR decomposition interally.

Supports methods:

• Sovler::Solver(SparseMatrix<T>& mat) construct from a matrix
• Vector<T> Sovler::solve(const Vector<T>& rhs) solve and return result in new vector
• void Sovler::solve(Vector<T>& result, const Vector<T>& rhs) solve and place result in existing vector
• size_t Sovler::rank() report the rank of the matrix. Some solvers may give only an approximate rank.

Warning: The Eigen built-in sparse QR solver is very inefficient for many problems. Also, it doesn’t work well for underdetermined systems.

template <typename<T>> class SquareSolver

Solve a system with a square matrix. Uses an LU decomposition interally.

Supports methods:

• SquareSovler::Solver(SparseMatrix<T>& mat) construct from a matrix
• Vector<T> SquareSovler::solve(const Vector<T>& rhs) solve and return result in new vector
• void SquareSovler::solve(Vector<T>& result, const Vector<T>& rhs) solve and place result in existing vector
template <typename<T>> class PositiveDefiniteSolver

Supports methods:

• PositiveDefiniteSolver::Solver(SparseMatrix<T>& mat) construct from a matrix
• Vector<T> PositiveDefiniteSolver::solve(const Vector<T>& rhs) solve and return result in new vector
• void PositiveDefiniteSolver::solve(Vector<T>& result, const Vector<T>& rhs) solve and place result in existing vector

Solve a system with a symmetric positive (semi-)definite matrix. Uses an LDLT decomposition interally.

## Eigenproblem solvers

These routines build on top of the direct solvers to solve eigenvalue problems using power methods.

Vector<T> smallestEigenvectorPositiveDefinite(SparseMatrix<T>& energyMatrix, SparseMatrix<T>& massMatrix, size_t nIterations = 50)

Solves the eigenvector problem $A x = \lambda M x$ for the smallest-eigenvalue’d nontrivial eigenvector $x$ of a positive definite sparse matrix $A$.

std::vector<Vector<T>> smallestKEigenvectorsPositiveDefinite(SparseMatrix<T>& energyMatrix, SparseMatrix<T>& massMatrix, size_t kEigenvalues, size_t nIterations = 50)

Solves the eigenvector problem $A x = \lambda M x$ for the first $k$ smallest-eigenvalue’d nontrivial eigenvectors $x$ of a positive definite sparse matrix $A$.

Vector<T> smallestEigenvectorSquare(SparseMatrix<T>& energyMatrix, SparseMatrix<T>& massMatrix, size_t nIterations = 50)

Solves the eigenvector problem $A x = \lambda M x$ for the smallest-eigenvalue’d nontrivial eigenvector $x$ of a square matrix $A$.

Vector<T> largestEigenvector(SparseMatrix<T>& energyMatrix, SparseMatrix<T>& massMatrix, size_t nIterations = 50)

Solves the eigenvector problem $A x = \lambda M x$ for the largest-eigenvalue’d nontrivial eigenvector $x$ of a square matrix $A$.

## Utilities

double residual(const SparseMatrix<T>& matrix, const Vector<T>& lhs, const Vector<T>& rhs)

Measure the L2 residual of a linear system as $||Ax - b||_2$.