math_m Module

This file provides a common interface to a large library of mathematical functions and subroutines. See the documentation of individual interfaces below for more information about the contents of the mathematical library.


Uses


Interfaces

public interface trace

Public interface for functions that calculate a matrix trace.

  • public pure function matrix_trace(A) result(r)

    Calculates the trace of a general complex matrix.

    Arguments

    Type IntentOptional Attributes Name
    complex(kind=wp), intent(in), dimension(:, :) :: A

    Matrix [n×m]

    Return Value complex(kind=wp)

    r = Tr(A)

public interface inverse

Public interface for functions that calculate a matrix inverse.

  • public pure function matrix_inverse_re(A) result(R)

    Wrapper for matrix_inverse_cx that operates on real matrices.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:, :) :: A

    Matrix A [n×n]

    Return Value real(kind=wp), dimension(size(A, 1), size(A, 1))

    Matrix R=A¯¹

  • public pure function matrix_inverse_cx(A) result(R)

    Inverts a square matrix via Gauss-Jordan elimination with partial pivot (general) or a cofactoring algorithm (2x2 matrices). The implementation is based on Algorithm #2 in "Efficient matrix inversion via Gauss-Jordan elimination and its parallelization" by E.S. Quintana et al. (1998).

    Arguments

    Type IntentOptional Attributes Name
    complex(kind=wp), intent(in), dimension(:, :) :: A

    Matrix A [n×n]

    Return Value complex(kind=wp), dimension(size(A, 1), size(A, 1))

    Matrix R=A¯¹

public interface diag

Public interface for functions that deal with matrix diagonals.

  • public pure function matrix_diag(A, B) result(R)

    Constructs a block-diagonal matrix R from two general matrices A and B.

    Arguments

    Type IntentOptional Attributes Name
    complex(kind=wp), intent(in), dimension(:, :) :: A

    Left matrix [n×m]
    complex(kind=wp), intent(in), dimension(:, :) :: B

    Right matrix [p×q]

    Return Value complex(kind=wp), dimension(size(A, 1) + size(B, 1), size(A, 2) + size(B, 2))

    R = Diag(A,B)

  • public pure function vector_diag(A) result(r)

    Extracts the diagonal of a general complex matrix.

    Arguments

    Type IntentOptional Attributes Name
    complex(kind=wp), intent(in), dimension(:, :) :: A

    Matrix [n×m]

    Return Value complex(kind=wp), dimension(min(size(A, 1), size(A, 2)))

    r = Diag(A)

public interface mean

Public interface for routines that calculate the mean value.

  • public pure function mean_array_re(x) result(r)

    Calculates the mean value of a real-valued array.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Real-valued array

    Return Value real(kind=wp)

    Mean value

  • public pure function mean_array_cx(x) result(r)

    Calculates the mean value of a complex-valued array.

    Arguments

    Type IntentOptional Attributes Name
    complex(kind=wp), intent(in), dimension(:) :: x

    Complex-valued array

    Return Value complex(kind=wp)

    Mean value

public interface differentiate

Public interface for various differentiation routines.

  • public pure function differentiate_array_re(x, y) result(r)

    Calculates the numerical derivative of an array y wrt. x using central differences at the interior points and forward/backward differences at the exterior points. All three approaches yield two-point approximations of the derivatives, thus the mesh spacing does not have to be uniform.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    real(kind=wp), intent(in), dimension(size(x)) :: y

    Function y(x)

    Return Value real(kind=wp), dimension(size(x))

    Derivative dy/dx

  • public pure function differentiate_array_cx(x, y) result(r)

    Complex version of differentiate_array_re.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    complex(kind=wp), intent(in), dimension(size(x)) :: y

    Function y(x)

    Return Value complex(kind=wp), dimension(size(x))

    Derivative dy/dx

public interface integrate

Public interface for various integration routines.

  • public pure function integrate_array_re(x, y) result(r)

    Calculates the integral of an array y wrt. x using the trapezoid method. The mesh spacing does not necessarily have to be uniform.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    real(kind=wp), intent(in), dimension(size(x)) :: y

    Function y(x)

    Return Value real(kind=wp)

    Integral ∫y(x)·dx

  • public pure function integrate_array_cx(x, y) result(r)

    Complex version of integrate_array_re.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    complex(kind=wp), intent(in), dimension(size(x)) :: y

    Function y(x)

    Return Value complex(kind=wp)

    Integral ∫y(x)·dx

  • public function integrate_range_re(x, y, a, b) result(r)

    Constructs a piecewise hermitian cubic interpolation of an array y(x) from discrete numerical data, and then integrates the interpolation in the range (a, b). The mesh spacing does not have to be uniform.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    real(kind=wp), intent(in), dimension(size(x)) :: y

    Function y(x)
    real(kind=wp), intent(in) :: a

    Left endpoint
    real(kind=wp), intent(in) :: b

    Right endpoint

    Return Value real(kind=wp)

    Integral ∫y(x)·dx

  • public function integrate_range_cx(x, y, a, b) result(r)

    Complex version of integrate_range_re.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    complex(kind=wp), intent(in), dimension(size(x)) :: y

    Function y(x)
    real(kind=wp), intent(in) :: a

    Left endpoint
    real(kind=wp), intent(in) :: b

    Right endpoint

    Return Value complex(kind=wp)

    Integral ∫y(x)·dx

public interface interpolate

Public interface for various interpolation routines.

  • public function interpolate_point_re(x, y, p) result(r)

    Wrapper for interpolate_array_re that accepts scalar arguments.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    real(kind=wp), intent(in), dimension(size(x)) :: y

    Function y(x)
    real(kind=wp), intent(in) :: p

    Single point p

    Return Value real(kind=wp)

    Interpolation y(p)

  • public function interpolate_point_cx(x, y, p) result(r)

    Complex version of interpolate_point_re.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    complex(kind=wp), intent(in), dimension(size(x)) :: y

    Function y(x)
    real(kind=wp), intent(in) :: p

    Single point p

    Return Value complex(kind=wp)

    Interpolation y(p)

  • public function interpolate_array_re(x, y, p) result(r)

    Constructs a piecewise hermitian cubic interpolation of an array y(x) based on discrete numerical data and evaluates the interpolation at p. Note that the mesh spacing does not necessarily have to be uniform.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    real(kind=wp), intent(in), dimension(size(x)) :: y

    Function y(x)
    real(kind=wp), intent(in), dimension(:) :: p

    Point array p

    Return Value real(kind=wp), dimension(size(p))

    Interpolation y(p)

  • public function interpolate_array_cx(x, y, p) result(r)

    Complex version of interpolate_array_re.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    complex(kind=wp), intent(in), dimension(size(x)) :: y

    Function y(x)
    real(kind=wp), intent(in), dimension(:) :: p

    Point array p

    Return Value complex(kind=wp), dimension(size(p))

    Interpolation y(p)

  • public pure function interpolate_point_matrix_re(x, y, p) result(r)

    Interpolates a matrix function using Catmull-Rom splines.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(in), dimension(:) :: x

    Variable x
    real(kind=wp), intent(in), dimension(:, :, :) :: y

    Function y(x)
    real(kind=wp), intent(in) :: p

    Single point p

    Return Value real(kind=wp), dimension(size(y, 1), size(y, 2))

    Interpolation y(p)

public interface linspace

Public interface for routines that initialize arrays.

  • public pure subroutine linspace_array_re(array, first, last)

    Populates an array with elements from first to last, inclusive.

    Arguments

    Type IntentOptional Attributes Name
    real(kind=wp), intent(out), dimension(:) :: array

    Output array to populate
    real(kind=wp), intent(in) :: first

    Value of first element
    real(kind=wp), intent(in) :: last

    Value of last element