This module contains all mathematics support routines. More...

Data Types
interface	coth
	Returns hyperbolic cotangent of argument. More...

interface	cross_product
	Calculates the vector cross product of two vectors. More...

interface	crossproduct
	Calculates the vector cross product of two vectors. More...

interface	d_cross_product
	Calculates the the vector cross product of A*B in C and the N derivatives, D_C, of the vector cross product given the derivatives D_A and D_B of A and B. More...

interface	dcrossproduct
	Calculates the the vector cross product of a x b in c and the n derivatives, dc, of the vector cross product given the derivatives da and db of a and b. More...

interface	determinant
	Returns the determinant of a matrix. More...

interface	edp
	Calculates the elliptic integral of the second kind - E(m). More...

interface	eigenvalue
	Returns the eigenvalues of a matrix. More...

interface	eigenvector
	Returns the eigenvectors of a matrix. More...

interface	i0
	Calculates the modified Bessel function of the first kind of order 0 using the approximation of Abromowitz and Stegun. More...

interface	i1
	Calculates the modified Bessel function of the first kind of order 1 using the approximation of Abromowitz and Stegun. More...

interface	identitymatrix
	Returns the identity matrix. More...

interface	invert
	Returns the inverse of a matrix. More...

interface	k0
	Calculates the modified Bessel function of the second kind of order 0 using the approximation of Abromowitz and Stegun. More...

interface	k1
	Calculates the modified Bessel function of the second kind of order 1 using the approximation of Abromowitz and Stegun. More...

interface	kdp
	Calculates the elliptic integral of the first kind - K(m). More...

interface	l2norm
	Returns the L2 norm of a vector. More...

interface	matrix_product
	Calculates and returns the matrix-product A*B in the matrix C. More...

interface	matrix_transpose
	Returns the transpose of a matrix A in A^T. More...

interface	matrix_vector_product
	Calculates and returns the matrix-vector-product A*b in the vector c. More...

interface	matrixproduct
	Calculates and returns the matrix-product A*B in the matrix C. More...

interface	matrixproducttranspose
	Calculates and returns the matrix-product-transpose A*B^T in the matrix C. More...

interface	matrixtranspose
	Returns the transpose of a matrix A in A^T. More...

interface	matrixtransposeproduct
	Calculates and returns the matrix-transpose product A^T*B in the matrix C. More...

interface	matrixtransposevectorproduct
	Calculates and returns the matrix-transpose vector-product A^T*b in the vector c. More...

interface	matrixvectorproduct
	Calculates and returns the matrix-vector-product A*b in the vector c. More...

interface	norm_cross_product
	Calculates the normalised vector cross product of two vectors. More...

interface	normalise
	Normalises a vector. More...

interface	normcrossproduct
	Calculates the normalised vector cross product of two vectors. More...

interface	solve_small_linear_system
	Solves a small linear system Ax=b. More...

interface	solvesmalllinearsystem
	Solves a small linear system Ax=b. More...

Functions/Subroutines
subroutine	crossproductintg (a, b, c, err, error,)
	Calculates and returns the vector cross-product of the integer vectors a x b in c. More...

subroutine	crossproductsp (a, b, c, err, error,)
	Calculates and returns the vector cross-product of the single precision vectors a x b in c. More...

subroutine	crossproductdp (a, b, c, err, error,)
	Calculates and returns the vector cross-product of the double precision vectors a x b in c. More...

subroutine	dcrossproductintg (n, a, b, c, da, db, dc, err, error,)
	Calculates the the vector cross product of a x b in c and the n derivatives, dc, of the vector cross product given the derivatives da and db of a and b for integer vectors. More...

subroutine	dcrossproductsp (n, a, b, c, da, db, dc, err, error,)
	Calculates the the vector cross product of a x b in c and the n derivatives, dc, of the vector cross product given the derivatives da and db of a and b for single precision vectors. More...

subroutine	dcrossproductdp (n, a, b, c, da, db, dc, err, error,)
	Calculates the the vector cross product of a x b in c and the n derivatives, dc, of the vector cross product given the derivatives da and db of a and b for double precision vectors. More...

subroutine	matrixvectorproductsp (A, b, c, err, error,)
	Calculates and returns the matrix-vector product of the single precision vector A*b in c. More...

subroutine	matrixvectorproductdp (A, b, c, err, error,)
	Calculates and returns the matrix-vector product of the double precision vectir A*b in c. More...

subroutine	matrixtransposevectorproductsp (A, b, c, err, error,)
	Calculates and returns the matrix-transpose vector product of the single precision vector A^T*b in c. More...

subroutine	matrixtransposevectorproductdp (A, b, c, err, error,)
	Calculates and returns the matrix-transpose vector product of the double precision vector A^T*b in c. More...

integer(intg) function	determinantfullintg (A, err, error)
	Returns the determinant of a full integer matrix A. More...

real(sp) function	determinantfullsp (A, err, error)
	Returns the determinant of a full single precision matrix A. More...

real(dp) function	determinantfulldp (A, err, error)
	Returns the determinant of a full double precision matrix A. More...

pure real(dp) function	edpdp (x)
	Calculates the elliptic integral of the second kind - E(m), for a double precision argument. More...

pure real(sp) function	edpsp (x)
	Calculates the elliptic integral of the second kind - E(m), for a single precision argument. More...

subroutine	eigenvaluefullsp (A, eValues, err, error,)
	Returns the eigenvalues of a full single precision matrix A. More...

subroutine	eigenvaluefulldp (A, eValues, err, error,)
	Returns the eigenvalues of a full double precision matrix A. More...

subroutine	eigenvectorfullsp (A, eValue, eVector, err, error,)
	Returns the normalised eigenvector of a full single precision symmetric matrix A that corresponds to the eigenvalue eValue. More...

subroutine	eigenvectorfulldp (A, eValue, eVector, err, error,)
	Returns the normalised eigenvector of a full double precision symmetric matrix A that corresponds to the eigenvalue eValue. More...

pure real(dp) function	i0dp (x)
	Calculates the modified Bessel function of the first kind of order 0 using the approximation of Abromowitz and Stegun, for a double precision argument. More...

pure real(sp) function	i0sp (x)
	Calculates the modified Bessel function of the first kind of order 0 using the approximation of Abromowitz and Stegun, for a single precision argument. More...

pure real(dp) function	i1dp (x)
	Calculates the modified Bessel function of the first kind of order 1 using the approximation of Abromowitz and Stegun, for a double precision argument. More...

pure real(sp) function	i1sp (x)
	Calculates the modified Bessel function of the first kind of order 1 using the approximation of Abromowitz and Stegun, for a single precision argument. More...

subroutine	identitymatrixsp (A, err, error,)
	Returns an identity matrix. More...

subroutine	identitymatrixdp (A, err, error,)
	Returns an identity matrix. More...

subroutine	invertfullsp (A, B, det, err, error,)
	Inverts a full single precision matrix A to give matrix B and returns the determinant of A in det. More...

subroutine	invertfulldp (A, B, det, err, error,)
	Inverts a full double precision matrix A to give matrix B and returns the determinant of A in det. More...

pure real(dp) function	k0dp (x)
	Calculates the modified Bessel function of the second kind of order 1 using the approximation of Abromowitz and Stegun, for a double precision argument. More...

pure real(sp) function	k0sp (x)
	Calculates the modified Bessel function of the second kind of order 0 using the approximation of Abromowitz and Stegun, for a single precision argument. More...

pure real(dp) function	k1dp (x)
	Calculates the modified Bessel function of the second kind of order 1 using the approximation of Abromowitz and Stegun, for a double precision argument. More...

pure real(sp) function	k1sp (x)
	Calculates the modified Bessel function of the second kind of order 1 using the approximation of Abromowitz and Stegun, for a single precision argument. More...

pure real(dp) function	kdpdp (x)
	Calculates the elliptic integral of the first kind - K(m), for a double precision argument. More...

pure real(sp) function	kdpsp (x)
	Calculates the elliptic integral of the first kind - K(m), for a single precision argument. More...

pure real(sp) function	l2normsp (a)
	Returns the L2-norm of the single precision vector a. More...

real(dp) function	l2normdp (A)
	Returns the L2-norm of the double precision vector a. More...

subroutine	matrixproductsp (A, B, C, err, error,)
	Calculates and returns the matrix-product of the single precision matrix A*B in C for single precision arguments. More...

subroutine	matrixproductdp (A, B, C, err, error,)
	Calculates and returns the matrix-product of the double precision matrix A*B in C. More...

subroutine	matrixtransposeproductsp (A, B, C, err, error,)
	Calculates and returns the matrix-transpose product of the single precision matrix A^T*B in C for single precision arguments. More...

subroutine	matrixtransposeproductdp (A, B, C, err, error,)
	Calculates and returns the matrix-transpose product of the double precision matrix A^T*B in C for double precision arguments. More...

subroutine	matrixproducttransposesp (A, B, C, err, error,)
	Calculates and returns the matrix-product-transpose of the single precision matrix A*B^T in C for single precision arguments. More...

subroutine	matrixproducttransposedp (A, B, C, err, error,)
	Calculates and returns the matrix-product-transpose of the double precision matrix A*B^T in C. More...

subroutine	matrixtransposesp (A, AT, err, error,)
	Returns the transpose of a single precision matrix A in AT. More...

subroutine	matrixtransposedp (A, AT, err, error,)
	Returns the transpose of a double precision matrix A in AT. More...

real(sp) function, dimension(size(a, 1))	normalisesp (a, err, error)
	Normalises a real single precision vector a. More...

real(dp) function, dimension(size(a, 1))	normalisedp (a, err, error)
	Normalises a real double precision vector a. More...

subroutine	normcrossproductsp (a, b, c, err, error,)
	Calculates and returns the normalised vector cross-prouct of the single precision vectors a x b in c. More...

subroutine	normcrossproductdp (a, b, c, err, error,)
	Calculates and returns the normalised vector cross-prouct of the double precision vectors a x b in c. More...

subroutine	solvesmalllinearsystemsp (A, x, b, err, error,)
	Finds the solution to a small single precision linear system Ax=b. More...

subroutine	solvesmalllinearsystemdp (A, x, b, err, error,)
	Finds the solution to a small double precision linear system Ax=b. More...

real(sp) function	cothsp (a)
	Calculates single precision hyperbolic cotangent function. More...

real(dp) function	cothdp (a)
	Calculates double precision hyperbolic cotangent function. More...

subroutine, public	spline_cubic_set (n, t, y, ibcbeg, ybcbeg, ibcend, ybcend, ypp, err, error,)
	Calculates second derivatives of a cubic spline function for a tabulated function y(x). Call spline_cubic_val to evaluate at t values. algorithm adapted from John Burkhardt's spline_cubic_set routine from the SPLINE package (http://people.sc.fsu.edu/~jburkardt/f_src/spline/spline.html) More...

subroutine, public	s3_fs (a1, a2, a3, n, b, x, err, error,)
	S3_FS factors and solves a tridiagonal linear system. algorithm adapted from John Burkhardt's s3_fs routine from the SPLINE package (http://people.sc.fsu.edu/~jburkardt/f_src/spline/spline.html) More...

subroutine, public	spline_cubic_val (n, t, y, ypp, tval, yval, ypval, yppval, err, error,)
	Evaluates a cubic spline at a specified point. First call spline_cubic_set to calculate derivatives algorithm adapted from John Burkhardt's spline_cubic_val routine from the SPLINE package (http://people.sc.fsu.edu/~jburkardt/f_src/spline/spline.html) More...

Detailed Description

This module contains all mathematics support routines.

Function/Subroutine Documentation

real(dp) function maths::cothdp ( real(dp), intent(in) a )

private

Calculates double precision hyperbolic cotangent function.

Parameters

[in] a argument to perform coth() on

Definition at line 2556 of file maths.f90.

real(sp) function maths::cothsp ( real(sp), intent(in) a )

private

Calculates single precision hyperbolic cotangent function.

Parameters

[in] a argument to perform coth() on

Definition at line 2538 of file maths.f90.

subroutine maths::crossproductdp	(	real(dp), dimension(:), intent(in)	a,
		real(dp), dimension(:), intent(in)	b,
		real(dp), dimension(:), intent(out)	c,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the vector cross-product of the double precision vectors a x b in c.

Parameters

[in]	a	The first vector in the cross product
[in]	b	The second vector in the cross product
[out]	c	On exit, the cross product of the first and second vectors
[out]	err	The error code
[out]	error	The error string

Definition at line 352 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::crossproductintg	(	integer(intg), dimension(:), intent(in)	a,
		integer(intg), dimension(:), intent(in)	b,
		integer(intg), dimension(:), intent(out)	c,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the vector cross-product of the integer vectors a x b in c.

Parameters

[in]	a	The first vector in the cross product
[in]	b	The second vector in the cross product
[out]	c	On exit, the cross product of the first and second vectors
[out]	err	The error code
[out]	error	The error string

Definition at line 270 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::crossproductsp	(	real(sp), dimension(:), intent(in)	a,
		real(sp), dimension(:), intent(in)	b,
		real(sp), dimension(:), intent(out)	c,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the vector cross-product of the single precision vectors a x b in c.

Parameters

[in]	a	The first vector in the cross product
[in]	b	The second vector in the cross product
[out]	c	On exit, the cross product of the first and second vectors
[out]	err	The error code
[out]	error	The error string

Definition at line 311 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::dcrossproductdp	(	integer(intg), intent(in)	n,
		real(dp), dimension(:), intent(in)	a,
		real(dp), dimension(:), intent(in)	b,
		real(dp), dimension(:), intent(out)	c,
		real(dp), dimension(:,:), intent(in)	da,
		real(dp), dimension(:,:), intent(in)	db,
		real(dp), dimension(:,:), intent(out)	dc,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates the the vector cross product of a x b in c and the n derivatives, dc, of the vector cross product given the derivatives da and db of a and b for double precision vectors.

Parameters

[in]	n	The number of derivatives
[in]	a	The a vector
[in]	b	The b vector
[out]	c	On exit, the cross product of a x b
[in]	da	The n derivatives of a
[in]	db	The n derivatives of b
[out]	dc	On exit, the derivatives of c
[out]	err	The error code
[out]	error	The error string

Definition at line 502 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::dcrossproductintg	(	integer(intg), intent(in)	n,
		integer(intg), dimension(:), intent(in)	a,
		integer(intg), dimension(:), intent(in)	b,
		integer(intg), dimension(:), intent(out)	c,
		integer(intg), dimension(:,:), intent(in)	da,
		integer(intg), dimension(:,:), intent(in)	db,
		integer(intg), dimension(:,:), intent(out)	dc,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates the the vector cross product of a x b in c and the n derivatives, dc, of the vector cross product given the derivatives da and db of a and b for integer vectors.

Parameters

[in]	n	The number of derivatives
[in]	a	The a vector
[in]	b	The b vector
[out]	c	On exit, the cross product of a x b
[in]	da	The n derivatives of a
[in]	db	The n derivatives of b
[out]	dc	On exit, the derivatives of c
[out]	err	The error code
[out]	error	The error string

Definition at line 394 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::dcrossproductsp	(	integer(intg), intent(in)	n,
		real(sp), dimension(:), intent(in)	a,
		real(sp), dimension(:), intent(in)	b,
		real(sp), dimension(:), intent(out)	c,
		real(sp), dimension(:,:), intent(in)	da,
		real(sp), dimension(:,:), intent(in)	db,
		real(sp), dimension(:,:), intent(out)	dc,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates the the vector cross product of a x b in c and the n derivatives, dc, of the vector cross product given the derivatives da and db of a and b for single precision vectors.

Parameters

[in]	n	The number of derivatives
[in]	a	The a vector
[in]	b	The b vector
[out]	c	On exit, the cross product of a x b
[in]	da	The n derivatives of a
[in]	db	The n derivatives of b
[out]	dc	On exit, the derivatives of c
[out]	err	The error code
[out]	error	The error string

Definition at line 448 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

real(dp) function maths::determinantfulldp	(	real(dp), dimension(:,:), intent(in)	A,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns the determinant of a full double precision matrix A.

Parameters

[in]	a	The matrix to find the determinant of
[out]	err	The error code
[out]	error	The error string

Definition at line 801 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

integer(intg) function maths::determinantfullintg	(	integer(intg), dimension(:,:), intent(in)	A,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns the determinant of a full integer matrix A.

Parameters

[in]	a	The matrix to find the determinant of
[out]	err	The error code
[out]	error	The error string

Definition at line 719 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

real(sp) function maths::determinantfullsp	(	real(sp), dimension(:,:), intent(in)	A,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns the determinant of a full single precision matrix A.

Parameters

[in]	a	The matrix to find the determinant of
[out]	err	The error code
[out]	error	The error string

Definition at line 760 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

pure real(dp) function maths::edpdp ( real(dp), intent(in) x )

private

Calculates the elliptic integral of the second kind - E(m), for a double precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 842 of file maths.f90.

pure real(sp) function maths::edpsp ( real(sp), intent(in) x )

private

Calculates the elliptic integral of the second kind - E(m), for a single precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 873 of file maths.f90.

subroutine maths::eigenvaluefulldp	(	real(dp), dimension(:,:), intent(in)	A,
		real(dp), dimension(:), intent(out)	eValues,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns the eigenvalues of a full double precision matrix A.

Parameters

[in]	a	The matrix to find the eigenvalues of
[out]	evalues	On exit, the eigenvalues of the matrix
[out]	err	The error code
[out]	error	The error string

Definition at line 993 of file maths.f90.

References base_routines::enters(), base_routines::exits(), constants::pi, and constants::zero_tolerance_dp.

subroutine maths::eigenvaluefullsp	(	real(sp), dimension(:,:), intent(in)	A,
		real(sp), dimension(:), intent(out)	eValues,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns the eigenvalues of a full single precision matrix A.

Parameters

[in]	a	The matrix to find the eignenvalues for
[out]	evalues	On exit, the eignevalues
[out]	err	The error code
[out]	error	The error string

Definition at line 905 of file maths.f90.

References base_routines::enters(), base_routines::exits(), and constants::zero_tolerance_sp.

subroutine maths::eigenvectorfulldp	(	real(dp), dimension(:,:), intent(in)	A,
		real(dp), intent(in)	eValue,
		real(dp), dimension(:), intent(out)	eVector,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns the normalised eigenvector of a full double precision symmetric matrix A that corresponds to the eigenvalue eValue.

Parameters

[in]	a	The matrix to find the eignevectors for
[in]	evalue	The eigenvalue to find the eignevector for
[out]	evector	On exit, the eigenvector corresponding the the eigenvalue
[out]	err	The error code
[out]	error	The error string

Definition at line 1168 of file maths.f90.

References base_routines::enters(), base_routines::exits(), and constants::zero_tolerance_dp.

subroutine maths::eigenvectorfullsp	(	real(sp), dimension(:,:), intent(in)	A,
		real(sp), intent(in)	eValue,
		real(sp), dimension(:), intent(out)	eVector,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns the normalised eigenvector of a full single precision symmetric matrix A that corresponds to the eigenvalue eValue.

Parameters

[in]	a	The matrix to find the eignevectors for
[in]	evalue	The eigenvalue to find the eignevector for
[out]	evector	On exit, the eigenvector corresponding the the eigenvalue
[out]	err	The error code
[out]	error	The error string

Definition at line 1081 of file maths.f90.

References base_routines::enters(), base_routines::exits(), and constants::zero_tolerance_sp.

pure real(dp) function maths::i0dp ( real(dp), intent(in) x )

private

Calculates the modified Bessel function of the first kind of order 0 using the approximation of Abromowitz and Stegun, for a double precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 1256 of file maths.f90.

pure real(sp) function maths::i0sp ( real(sp), intent(in) x )

private

Calculates the modified Bessel function of the first kind of order 0 using the approximation of Abromowitz and Stegun, for a single precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 1287 of file maths.f90.

pure real(dp) function maths::i1dp ( real(dp), intent(in) x )

private

Calculates the modified Bessel function of the first kind of order 1 using the approximation of Abromowitz and Stegun, for a double precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 1318 of file maths.f90.

pure real(sp) function maths::i1sp ( real(sp), intent(in) x )

private

Calculates the modified Bessel function of the first kind of order 1 using the approximation of Abromowitz and Stegun, for a single precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 1349 of file maths.f90.

subroutine maths::identitymatrixdp	(	real(dp), dimension(:,:), intent(out)	A,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns an identity matrix.

Parameters

[out]	a	On exit, the identity matrix
[out]	err	The error code
[out]	error	The error string

Definition at line 1431 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::identitymatrixsp	(	real(sp), dimension(:,:), intent(out)	A,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns an identity matrix.

Parameters

[out]	a	On exit, the identity matrix
[out]	err	The error code
[out]	error	The error string

Definition at line 1379 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::invertfulldp	(	real(dp), dimension(:,:), intent(in)	A,
		real(dp), dimension(:,:), intent(out)	B,
		real(dp), intent(out)	det,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Inverts a full double precision matrix A to give matrix B and returns the determinant of A in det.

Parameters

[in]	a	The matrix A to invert
[out]	b	On exit, the inverse of A
[out]	det	On exit, the determinant of A
[out]	err	The error code
[out]	error	The error string

Definition at line 1556 of file maths.f90.

References base_routines::enters(), base_routines::exits(), and constants::zero_tolerance_dp.

subroutine maths::invertfullsp	(	real(sp), dimension(:,:), intent(in)	A,
		real(sp), dimension(:,:), intent(out)	B,
		real(sp), intent(out)	det,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Inverts a full single precision matrix A to give matrix B and returns the determinant of A in det.

Parameters

[in]	a	The A matrix to invert
[out]	b	On exit, the inverse of A
[out]	det	On exit, the determinant of A
[out]	err	The error code
[out]	error	The error string

Definition at line 1483 of file maths.f90.

References base_routines::enters(), base_routines::exits(), and constants::zero_tolerance_sp.

pure real(dp) function maths::k0dp ( real(dp), intent(in) x )

private

Calculates the modified Bessel function of the second kind of order 1 using the approximation of Abromowitz and Stegun, for a double precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 1630 of file maths.f90.

pure real(sp) function maths::k0sp ( real(sp), intent(in) x )

private

Calculates the modified Bessel function of the second kind of order 0 using the approximation of Abromowitz and Stegun, for a single precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 1677 of file maths.f90.

pure real(dp) function maths::k1dp ( real(dp), intent(in) x )

private

Calculates the modified Bessel function of the second kind of order 1 using the approximation of Abromowitz and Stegun, for a double precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 1724 of file maths.f90.

pure real(sp) function maths::k1sp ( real(sp), intent(in) x )

private

Calculates the modified Bessel function of the second kind of order 1 using the approximation of Abromowitz and Stegun, for a single precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 1772 of file maths.f90.

pure real(dp) function maths::kdpdp ( real(dp), intent(in) x )

private

Calculates the elliptic integral of the first kind - K(m), for a double precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 1819 of file maths.f90.

pure real(sp) function maths::kdpsp ( real(sp), intent(in) x )

private

Calculates the elliptic integral of the first kind - K(m), for a single precision argument.

Parameters

[in] x The value to evaluate the function at

Definition at line 1853 of file maths.f90.

real(dp) function maths::l2normdp ( real(dp), dimension(:), intent(in) A )

private

Returns the L2-norm of the double precision vector a.

Parameters

[in] a The vector to calculate the L2 norm of

Definition at line 1908 of file maths.f90.

pure real(sp) function maths::l2normsp ( real(sp), dimension(:), intent(in) a )

private

Returns the L2-norm of the single precision vector a.

Parameters

[in] a The vector to calculate the L2 norm of

Definition at line 1887 of file maths.f90.

subroutine maths::matrixproductdp	(	real(dp), dimension(:,:), intent(in)	A,
		real(dp), dimension(:,:), intent(in)	B,
		real(dp), dimension(:,:), intent(out)	C,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the matrix-product of the double precision matrix A*B in C.

Parameters

[in]	a	The A matrix
[in]	b	The B matrix
[out]	c	On exit, the product matrix C=A*B
[out]	err	The error code
[out]	error	The error string

Definition at line 1979 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixproductsp	(	real(sp), dimension(:,:), intent(in)	A,
		real(sp), dimension(:,:), intent(in)	B,
		real(sp), dimension(:,:), intent(out)	C,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the matrix-product of the single precision matrix A*B in C for single precision arguments.

Parameters

[in]	a	The first matrix A
[in]	b	The second matrix B
[out]	c	On exit, the product matrix C=A*B
[out]	err	The error code
[out]	error	The error string

Definition at line 1929 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixproducttransposedp	(	real(dp), dimension(:,:), intent(in)	A,
		real(dp), dimension(:,:), intent(in)	B,
		real(dp), dimension(:,:), intent(out)	C,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the matrix-product-transpose of the double precision matrix A*B^T in C.

Parameters

[in]	a	The A matrix
[in]	b	The B matrix
[out]	c	On exit, the product matrix C=A*B^T
[out]	err	The error code
[out]	error	The error string

Definition at line 2178 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixproducttransposesp	(	real(sp), dimension(:,:), intent(in)	A,
		real(sp), dimension(:,:), intent(in)	B,
		real(sp), dimension(:,:), intent(out)	C,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the matrix-product-transpose of the single precision matrix A*B^T in C for single precision arguments.

Parameters

[in]	a	The first matrix A
[in]	b	The second matrix B
[out]	c	On exit, the product matrix C=A*B^T
[out]	err	The error code
[out]	error	The error string

Definition at line 2128 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixtransposedp	(	real(dp), dimension(:,:), intent(in)	A,
		real(dp), dimension(:,:), intent(out)	AT,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns the transpose of a double precision matrix A in AT.

Parameters

[in]	a	The matrix to take the transpose of
[out]	at	On exit, the transpose of the matrix
[out]	err	The error code
[out]	error	The error string

Definition at line 2277 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixtransposeproductdp	(	real(dp), dimension(:,:), intent(in)	A,
		real(dp), dimension(:,:), intent(in)	B,
		real(dp), dimension(:,:), intent(out)	C,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the matrix-transpose product of the double precision matrix A^T*B in C for double precision arguments.

Parameters

[in]	a	The first matrix A
[in]	b	The second matrix B
[out]	c	On exit, the product matrix C=A^T*B
[out]	err	The error code
[out]	error	The error string

Definition at line 2079 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixtransposeproductsp	(	real(sp), dimension(:,:), intent(in)	A,
		real(sp), dimension(:,:), intent(in)	B,
		real(sp), dimension(:,:), intent(out)	C,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the matrix-transpose product of the single precision matrix A^T*B in C for single precision arguments.

Parameters

[in]	a	The first matrix A
[in]	b	The second matrix B
[out]	c	On exit, the product matrix C=A^T*B
[out]	err	The error code
[out]	error	The error string

Definition at line 2029 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixtransposesp	(	real(sp), dimension(:,:), intent(in)	A,
		real(sp), dimension(:,:), intent(out)	AT,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Returns the transpose of a single precision matrix A in AT.

Parameters

[in]	a	The matrix to take the transpose of
[out]	at	On exit, the transpose of the matrix
[out]	err	The error code
[out]	error	The error string

Definition at line 2228 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixtransposevectorproductdp	(	real(dp), dimension(:,:), intent(in)	A,
		real(dp), dimension(:), intent(in)	b,
		real(dp), dimension(:), intent(out)	c,
		integer(intg)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the matrix-transpose vector product of the double precision vector A^T*b in c.

Parameters

[in]	a	The A matrix
[in]	b	The b vector
[out]	c	On exit, the product vector c=A^T*b
	err	The error code
[out]	error	The error string

Definition at line 678 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixtransposevectorproductsp	(	real(sp), dimension(:,:), intent(in)	A,
		real(sp), dimension(:), intent(in)	b,
		real(sp), dimension(:), intent(out)	c,
		integer(intg)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the matrix-transpose vector product of the single precision vector A^T*b in c.

Parameters

[in]	a	The A matrix
[in]	b	The b vector
[out]	c	On exit, the product vector c=A^T*b
	err	The error code
[out]	error	The error string

Definition at line 637 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixvectorproductdp	(	real(dp), dimension(:,:), intent(in)	A,
		real(dp), dimension(:), intent(in)	b,
		real(dp), dimension(:), intent(out)	c,
		integer(intg)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the matrix-vector product of the double precision vectir A*b in c.

Parameters

[in]	a	The A matrix
[in]	b	The b vector
[out]	c	On exit, the product vector c=A*b
	err	The error code
[out]	error	The error string

Definition at line 596 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::matrixvectorproductsp	(	real(sp), dimension(:,:), intent(in)	A,
		real(sp), dimension(:), intent(in)	b,
		real(sp), dimension(:), intent(out)	c,
		integer(intg)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the matrix-vector product of the single precision vector A*b in c.

Parameters

[in]	a	The A matrix
[in]	b	The b vector
[out]	c	On exit, the product vector c=A*b
	err	The error code
[out]	error	The error string

Definition at line 555 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

real(dp) function, dimension(size(a,1)) maths::normalisedp	(	real(dp), dimension(:), intent(in)	a,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Normalises a real double precision vector a.

Parameters

[in]	a	The vector to normalise
[out]	err	The error code
[out]	error	The error string

Definition at line 2359 of file maths.f90.

References base_routines::enters(), base_routines::exits(), and constants::zero_tolerance_dp.

real(sp) function, dimension(size(a,1)) maths::normalisesp	(	real(sp), dimension(:), intent(in)	a,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Normalises a real single precision vector a.

Parameters

[in]	a	The vector to normalise
[out]	err	The error code
[out]	error	The error string

Definition at line 2326 of file maths.f90.

References base_routines::enters(), base_routines::exits(), and constants::zero_tolerance_sp.

subroutine maths::normcrossproductdp	(	real(dp), dimension(:), intent(in)	a,
		real(dp), dimension(:), intent(in)	b,
		real(dp), dimension(:), intent(out)	c,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the normalised vector cross-prouct of the double precision vectors a x b in c.

Parameters

[in]	a	The first vector in the cross product
[in]	b	The second vector in the cross product
[out]	c	On exit, the normalised cross product of the first and second vectors
[out]	err	The error code
[out]	error	The error string

Definition at line 2420 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::normcrossproductsp	(	real(sp), dimension(:), intent(in)	a,
		real(sp), dimension(:), intent(in)	b,
		real(sp), dimension(:), intent(out)	c,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Calculates and returns the normalised vector cross-prouct of the single precision vectors a x b in c.

Parameters

[in]	a	The first vector in the cross product
[in]	b	The second vector in the cross product
[out]	c	On exit, the normalised cross product of the first and second vectors
[out]	err	The error code
[out]	error	The error string

Definition at line 2392 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine, public maths::s3_fs	(	real(dp), dimension(2:n), intent(inout)	a1,
		real(dp), dimension(1:n), intent(inout)	a2,
		real(dp), dimension(1:n-1), intent(inout)	a3,
		integer(intg), intent(in)	n,
		real(dp), dimension(n), intent(inout)	b,
		real(dp), dimension(n), intent(out)	x,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

S3_FS factors and solves a tridiagonal linear system. algorithm adapted from John Burkhardt's s3_fs routine from the SPLINE package (http://people.sc.fsu.edu/~jburkardt/f_src/spline/spline.html)

Parameters

[in,out]	a1	IN: nonzero diagonal of linear system OUT: factorization info
[in,out]	a2	IN: nonzero diagonal of linear system OUT: factorization info
[in,out]	a3	IN: nonzero diagonal of linear system OUT: factorization info
[in]	n	size of x,y arrays to interpolate values from
[in,out]	b	IN: RHS of linear system OUT: factorization info
[out]	x	solution of linear system
[out]	err	The error code
[out]	error	The error string

Definition at line 2678 of file maths.f90.

References base_routines::enters(), base_routines::exits(), and constants::zero_tolerance.

Referenced by spline_cubic_set().

subroutine maths::solvesmalllinearsystemdp	(	real(dp), dimension(:,:), intent(in)	A,
		real(dp), dimension(:), intent(out)	x,
		real(dp), dimension(:), intent(in)	b,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Finds the solution to a small double precision linear system Ax=b.

Parameters

[in]	a	The A matrix
[out]	x	On exit, the solution vector x
[in]	b	The RHS vector b
[out]	err	The error code
[out]	error	The error string

Definition at line 2493 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine maths::solvesmalllinearsystemsp	(	real(sp), dimension(:,:), intent(in)	A,
		real(sp), dimension(:), intent(out)	x,
		real(sp), dimension(:), intent(in)	b,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

private

Finds the solution to a small single precision linear system Ax=b.

Parameters

[in]	a	The A matrix
[out]	x	On exit, the solution vector x
[in]	b	The RHS vector b
[out]	err	The error code
[out]	error	The error string

Definition at line 2448 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

subroutine, public maths::spline_cubic_set	(	integer(intg), intent(in)	n,
		real(dp), dimension(n), intent(in)	t,
		real(dp), dimension(n), intent(in)	y,
		integer(intg), intent(in)	ibcbeg,
		real(dp), intent(in)	ybcbeg,
		integer(intg), intent(in)	ibcend,
		real(dp), intent(in)	ybcend,
		real(dp), dimension(n), intent(out)	ypp,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

Calculates second derivatives of a cubic spline function for a tabulated function y(x). Call spline_cubic_val to evaluate at t values. algorithm adapted from John Burkhardt's spline_cubic_set routine from the SPLINE package (http://people.sc.fsu.edu/~jburkardt/f_src/spline/spline.html)

Parameters

[in]	n	size of x,y arrays to interpolate values from
[in]	t	t array: known values
[in]	y	y array: values to interpolate
[in]	ibcbeg	left boundary condition flag
[in]	ybcbeg	1st derivative interpolating function at point 1 (left boundary)
[in]	ibcend	right boundary condition flag
[in]	ybcend	1st derivative interpolating function at point n (right boundary)
[out]	ypp	2nd derivatives of interpolating function at x values
[out]	err	The error code
[out]	error	The error string

Definition at line 2575 of file maths.f90.

References base_routines::enters(), base_routines::exits(), and s3_fs().

Referenced by navier_stokes_equations_routines::NavierStokes_PreSolveUpdateBoundaryConditions::navierstokes_presolveupdateboundaryconditions().

subroutine, public maths::spline_cubic_val	(	integer(intg), intent(in)	n,
		real(dp), dimension(n), intent(in)	t,
		real(dp), dimension(n), intent(in)	y,
		real(dp), dimension(n), intent(in)	ypp,
		real(dp), intent(in)	tval,
		real(dp), intent(out)	yval,
		real(dp), intent(out)	ypval,
		real(dp), intent(out)	yppval,
		integer(intg), intent(out)	err,
		type(varying_string), intent(out)	error
	)

Evaluates a cubic spline at a specified point. First call spline_cubic_set to calculate derivatives algorithm adapted from John Burkhardt's spline_cubic_val routine from the SPLINE package (http://people.sc.fsu.edu/~jburkardt/f_src/spline/spline.html)

Parameters

[in]	n	size of t,y arrays to interpolate values from
[in]	t	t array: known knot values
[in]	y	y array: data values to interpolate at the knots
[in]	ypp	2nd derivatives of interpolating function at t values
[in]	tval	point in t at which spline is to be evaluated
[out]	yval	spline interpolated y value at tval
[out]	ypval	first derivative of spline interpolated y value at tval
[out]	yppval	second derivative of spline interpolated y value at tval
[out]	err	The error code
[out]	error	The error string

Definition at line 2730 of file maths.f90.

References base_routines::enters(), and base_routines::exits().

Referenced by navier_stokes_equations_routines::NavierStokes_PreSolveUpdateBoundaryConditions::navierstokes_presolveupdateboundaryconditions().

Data Types

Functions/Subroutines

Detailed Description

Function/Subroutine Documentation