math_m Module

public pure function matrix_trace(A) result(r)

Calculate the trace of a general complex matrix.

Arguments

Return Value complex(kind=wp)

r = Tr(A)

public pure function matrix_inverse_re(A) result(R)

Wrapper for matrix_inverse_cx that allows the procedure to be used for real matrices.

Arguments

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)

Invert a square n×n matrix using Gauss-Jordan elimination with partial pivoting. In the special case n=2, the inverse is evaluated using a cofactoring algorithm. [This 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

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

Matrix R=A¯¹

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

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

Arguments

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)

Extract the diagonal of a general complex matrix.

Arguments

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

r = Diag(A)

public pure function mean_array_re(x) result(r)

Calculates the mean value of a real-valued array.

Arguments

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

Return Value complex(kind=wp)

Mean value

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

This function calculates the numerical derivative of an array y with respect to x, using a central difference approximation at the interior points and forward/backward difference approximations at the exterior points. Note that since all the three approaches yield two-point approximations of the derivative, the mesh spacing of x does not necessarily have to be uniform.

Arguments

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

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

Derivative dy/dx

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

This function calculates the integral of an array y with respect to x using a trapezoid approximation. Note that the mesh spacing of x does not necessarily have to be uniform.

Arguments

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

Return Value complex(kind=wp)

Integral ∫y(x)·dx

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

This function constructs a piecewise hermitian cubic interpolation of an array y(x) based on discrete numerical data, and subsequently evaluates the integral of the interpolation in the range (a,b). Note that the mesh spacing of x does not necessarily have to be uniform.

Arguments

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

Return Value complex(kind=wp)

Integral ∫y(x)·dx

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

Wrapper for interpolate_array_re that accepts scalar arguments.

Arguments

Return Value real(kind=wp)

Interpolation result y(p)

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

Complex version of interpolate_point_re.

Arguments

Return Value complex(kind=wp)

Interpolation result y(p)

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

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

Arguments

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

Interpolation result y(p)

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

Complex version of interpolate_array_re.

Arguments

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

Interpolation result y(p)

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

Perform a Piecewise Cubic Hermitian Interpolation of a matrix function using Catmull-Rom splines.

Arguments

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

Interpolation result y(p)

public pure subroutine linspace_array_re(array, first, last)

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

Arguments