Classes
Matrix functions module
Unsupported modules

This module aims to provide various methods for the computation of matrix functions. More...

Classes

class  MatrixExponential< MatrixType >
 Class for computing the matrix exponential. More...
struct  MatrixExponentialReturnValue< Derived >
 Proxy for the matrix exponential of some matrix (expression). More...
class  MatrixFunction< MatrixType, IsComplex >
 Class for computing matrix exponentials. More...
class  MatrixFunction< MatrixType, 0 >
 Partial specialization of MatrixFunction for real matrices. More...
class  MatrixFunction< MatrixType, 1 >
 Partial specialization of MatrixFunction for complex matrices. More...
class  MatrixFunctionAtomic< MatrixType >
 Helper class for computing matrix functions of atomic matrices. More...
class  MatrixFunctionReturnValue< Derived >
 Proxy for the matrix function of some matrix (expression). More...
class  StdStemFunctions< Scalar >
 Stem functions corresponding to standard mathematical functions. More...

Detailed Description

This module aims to provide various methods for the computation of matrix functions.

To use this module, add 

 #include <unsupported/Eigen/MatrixFunctions>

at the start of your source file.

This module defines the following MatrixBase methods.

These methods are the main entry points to this module.

Matrix functions are defined as follows. Suppose that $ f $ is an entire function (that is, a function on the complex plane that is everywhere complex differentiable). Then its Taylor series

\[ f(0) + f'(0) x + \frac{f''(0)}{2} x^2 + \frac{f'''(0)}{3!} x^3 + \cdots \]

converges to $ f(x) $. In this case, we can define the matrix function by the same series:

\[ f(M) = f(0) + f'(0) M + \frac{f''(0)}{2} M^2 + \frac{f'''(0)}{3!} M^3 + \cdots \]

MatrixBase methods defined in the MatrixFunctions module

The remainder of the page documents the following MatrixBase methods which are defined in the MatrixFunctions module.

MatrixBase::cos()

Compute the matrix cosine.

const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cos() const
Parameters:
[in]Ma square matrix.
Returns:
expression representing $ \cos(M) $.

This function calls matrixFunction() with StdStemFunctions::cos().

See also:
sin() for an example.

MatrixBase::cosh()

Compute the matrix hyberbolic cosine.

const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cosh() const
Parameters:
[in]Ma square matrix.
Returns:
expression representing $ \cosh(M) $

This function calls matrixFunction() with StdStemFunctions::cosh().

See also:
sinh() for an example.

MatrixBase::exp()

Compute the matrix exponential.

const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
Parameters:
[in]Mmatrix whose exponential is to be computed.
Returns:
expression representing the matrix exponential of M.

The matrix exponential of $ M $ is defined by

\[ \exp(M) = \sum_{k=0}^\infty \frac{M^k}{k!}. \]

The matrix exponential can be used to solve linear ordinary differential equations: the solution of $ y' = My $ with the initial condition $ y(0) = y_0 $ is given by $ y(t) = \exp(M) y_0 $.

The cost of the computation is approximately $ 20 n^3 $ for matrices of size $ n $. The number 20 depends weakly on the norm of the matrix.

The matrix exponential is computed using the scaling-and-squaring method combined with Padé approximation. The matrix is first rescaled, then the exponential of the reduced matrix is computed approximant, and then the rescaling is undone by repeated squaring. The degree of the Padé approximant is chosen such that the approximation error is less than the round-off error. However, errors may accumulate during the squaring phase.

Details of the algorithm can be found in: Nicholas J. Higham, "The scaling and squaring method for the matrix exponential revisited," SIAM J. Matrix Anal. Applic., 26:1179–1193, 2005.

Example: The following program checks that

\[ \exp \left[ \begin{array}{ccc} 0 & \frac14\pi & 0 \\ -\frac14\pi & 0 & 0 \\ 0 & 0 & 0 \end{array} \right] = \left[ \begin{array}{ccc} \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\ \frac12\sqrt2 & \frac12\sqrt2 & 0 \\ 0 & 0 & 1 \end{array} \right]. \]

This corresponds to a rotation of $ \frac14\pi $ radians around the z-axis.

#include <unsupported/Eigen/MatrixFunctions>
#include <iostream>

using namespace Eigen;

int main()
{
  const double pi = std::acos(-1.0);

  MatrixXd A(3,3);
  A << 0,    -pi/4, 0,
       pi/4, 0,     0,
       0,    0,     0;
  std::cout << "The matrix A is:\n" << A << "\n\n";
  std::cout << "The matrix exponential of A is:\n" << A.exp() << "\n\n";
}

Output: 

The matrix A is:
        0 -0.785398         0
 0.785398         0         0
        0         0         0

The matrix exponential of A is:
 0.707107 -0.707107         0
 0.707107  0.707107         0
        0         0         1
Note:
M has to be a matrix of float, double, complex<float> or complex<double> .

MatrixBase::matrixFunction()

Compute a matrix function.

const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::matrixFunction(typename internal::stem_function<typename internal::traits<Derived>::Scalar>::type f) const
Parameters:
[in]Margument of matrix function, should be a square matrix.
[in]fan entire function; f(x,n) should compute the n-th derivative of f at x.
Returns:
expression representing f applied to M.

Suppose that M is a matrix whose entries have type Scalar. Then, the second argument, f, should be a function with prototype 

ComplexScalar f(ComplexScalar, int)

where ComplexScalar = std::complex<Scalar> if Scalar is real (e.g., float or double) and ComplexScalar = Scalar if Scalar is complex. The return value of f(x,n) should be $ f^{(n)}(x) $, the n-th derivative of f at x.

This routine uses the algorithm described in: Philip Davies and Nicholas J. Higham, "A Schur-Parlett algorithm for computing matrix functions", SIAM J. Matrix Anal. Applic., 25:464–485, 2003.

The actual work is done by the MatrixFunction class.

Example: The following program checks that

\[ \exp \left[ \begin{array}{ccc} 0 & \frac14\pi & 0 \\ -\frac14\pi & 0 & 0 \\ 0 & 0 & 0 \end{array} \right] = \left[ \begin{array}{ccc} \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\ \frac12\sqrt2 & \frac12\sqrt2 & 0 \\ 0 & 0 & 1 \end{array} \right]. \]

This corresponds to a rotation of $ \frac14\pi $ radians around the z-axis. This is the same example as used in the documentation of exp().

#include <unsupported/Eigen/MatrixFunctions>
#include <iostream>

using namespace Eigen;

std::complex<double> expfn(std::complex<double> x, int)
{
  return std::exp(x);
}

int main()
{
  const double pi = std::acos(-1.0);

  MatrixXd A(3,3);
  A << 0,    -pi/4, 0,
       pi/4, 0,     0,
       0,    0,     0;

  std::cout << "The matrix A is:\n" << A << "\n\n";
  std::cout << "The matrix exponential of A is:\n" 
            << A.matrixFunction(expfn) << "\n\n";
}

Output: 

The matrix A is:
        0 -0.785398         0
 0.785398         0         0
        0         0         0

The matrix exponential of A is:
 0.707107 -0.707107         0
 0.707107  0.707107         0
        0         0         1

Note that the function expfn is defined for complex numbers x, even though the matrix A is over the reals. Instead of expfn, we could also have used StdStemFunctions::exp: 

A.matrixFunction(StdStemFunctions<std::complex<double> >::exp, &B);

MatrixBase::sin()

Compute the matrix sine.

const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sin() const
Parameters:
[in]Ma square matrix.
Returns:
expression representing $ \sin(M) $.

This function calls matrixFunction() with StdStemFunctions::sin().

Example: 

#include <unsupported/Eigen/MatrixFunctions>
#include <iostream>

using namespace Eigen;

int main()
{
  MatrixXd A = MatrixXd::Random(3,3);
  std::cout << "A = \n" << A << "\n\n";

  MatrixXd sinA = A.sin();
  std::cout << "sin(A) = \n" << sinA << "\n\n";

  MatrixXd cosA = A.cos();
  std::cout << "cos(A) = \n" << cosA << "\n\n";
  
  // The matrix functions satisfy sin^2(A) + cos^2(A) = I, 
  // like the scalar functions.
  std::cout << "sin^2(A) + cos^2(A) = \n" << sinA*sinA + cosA*cosA << "\n\n";
}

Output: 

A = 
 0.680375   0.59688 -0.329554
-0.211234  0.823295  0.536459
 0.566198 -0.604897 -0.444451

sin(A) = 
 0.679919    0.4579 -0.400612
-0.227278  0.821913    0.5358
 0.570141 -0.676728 -0.462398

cos(A) = 
 0.927728 -0.530361 -0.110482
0.00969246  0.889022 -0.137604
-0.132574  -0.04289   1.16475

sin^2(A) + cos^2(A) = 
           1            0    4.996e-16
-4.44089e-16            1  2.77556e-16
 7.21645e-16 -5.96745e-16            1

const MatrixBase::sinh()

Compute the matrix hyperbolic sine.

MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sinh() const
Parameters:
[in]Ma square matrix.
Returns:
expression representing $ \sinh(M) $

This function calls matrixFunction() with StdStemFunctions::sinh().

Example: 

#include <unsupported/Eigen/MatrixFunctions>
#include <iostream>

using namespace Eigen;

int main()
{
  MatrixXf A = MatrixXf::Random(3,3);
  std::cout << "A = \n" << A << "\n\n";

  MatrixXf sinhA = A.sinh();
  std::cout << "sinh(A) = \n" << sinhA << "\n\n";

  MatrixXf coshA = A.cosh();
  std::cout << "cosh(A) = \n" << coshA << "\n\n";
  
  // The matrix functions satisfy cosh^2(A) - sinh^2(A) = I, 
  // like the scalar functions.
  std::cout << "cosh^2(A) - sinh^2(A) = \n" << coshA*coshA - sinhA*sinhA << "\n\n";
}

Output: 

A = 
 0.680375   0.59688 -0.329554
-0.211234  0.823295  0.536459
 0.566198 -0.604897 -0.444451

sinh(A) = 
 0.682534  0.739989 -0.256871
-0.194928  0.826513  0.537546
 0.562584  -0.53163 -0.425199

cosh(A) = 
  1.07817  0.567069  0.132125
-0.00418609   1.11649  0.135361
 0.128891 0.0659991  0.851201

cosh^2(A) - sinh^2(A) = 
          1 7.15256e-07 5.96046e-08
1.95578e-07           1           0
-2.98023e-08 1.63913e-07           1
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