fstats_special_functions

public pure elemental function beta(a, b) result(rst)

Computes the beta function.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in)			::	a	The first argument of the function.
real(kind=real64),	intent(in)			::	b	The second argument of the function.

Return Value real(kind=real64)

The value of the beta function at $a$ and $b$ .

public pure elemental function digamma(x) result(rst)

Computes the digamma function.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in)			::	x	The value at which to evaluate the function.

Return Value real(kind=real64)

The function value.

public pure elemental function incomplete_beta(a, b, x) result(rst)

Computes the incomplete beta function.

Arguments

Type	Intent		Name
real(kind=real64),	intent(in)	::	a	The first argument of the function.
real(kind=real64),	intent(in)	::	b	The second argument of the function.
real(kind=real64),	intent(in)	::	x	The upper limit of the integration.

Return Value real(kind=real64)

The value of the incomplete beta function.

public pure elemental function incomplete_gamma_lower(a, x) result(rst)

Computes the lower incomplete gamma function.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in)			::	a	The coefficient value.
real(kind=real64),	intent(in)			::	x	The value at which to evaluate the function.

Return Value real(kind=real64)

The function value.

public pure elemental function incomplete_gamma_upper(a, x) result(rst)

Computes the upper incomplete gamma function.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in)			::	a	The coefficient value.
real(kind=real64),	intent(in)			::	x	The value at which to evaluate the function.

Return Value real(kind=real64)

The function value.

public pure elemental function log_regularized_beta(a, b, x, max_iterations) result(rst)

Log regularized beta; invalid inputs or nonconvergence return NaN. The leading factor and complementary branch are evaluated in log space.

Arguments

Type	Intent	Optional		Name
real(kind=real64),	intent(in)		::	a	Finite, positive beta shape parameters.
real(kind=real64),	intent(in)		::	b	Finite, positive beta shape parameters.
real(kind=real64),	intent(in)		::	x	Integration limit on [0, 1].
integer(kind=int32),	intent(in),	optional	::	max_iterations	Continued-fraction iteration limit; default 1000.

Return Value real(kind=real64)

Log ratio; x = 0 gives -infinity, x = 1 gives zero.

public pure elemental function log_regularized_gamma_lower(a, x, max_iterations) result(rst)

Log lower gamma ratio; invalid inputs or nonconvergence return NaN.

Arguments

Type	Intent	Optional		Name
real(kind=real64),	intent(in)		::	a	Finite, positive gamma shape parameter.
real(kind=real64),	intent(in)		::	x	Nonnegative integration limit; +infinity is accepted.
integer(kind=int32),	intent(in),	optional	::	max_iterations	Series/continued-fraction iteration limit; default 10000.

Return Value real(kind=real64)

Log P(a, x); x = 0 gives -infinity, x = +infinity gives zero.

public pure elemental function log_regularized_gamma_upper(a, x, max_iterations) result(rst)

Log upper gamma ratio; invalid inputs or nonconvergence return NaN.

Arguments

Type	Intent	Optional		Name
real(kind=real64),	intent(in)		::	a	Finite, positive gamma shape parameter.
real(kind=real64),	intent(in)		::	x	Nonnegative integration limit; +infinity is accepted.
integer(kind=int32),	intent(in),	optional	::	max_iterations	Series/continued-fraction iteration limit; default 10000.

Return Value real(kind=real64)

Log Q(a, x); x = 0 gives zero, x = +infinity gives -infinity.

public pure elemental function regularized_beta(a, b, x, max_iterations) result(rst)

Computes the regularized beta function.

Arguments

Type	Intent	Optional		Name
real(kind=real64),	intent(in)		::	a	The first argument of the function.
real(kind=real64),	intent(in)		::	b	The second argument of the function.
real(kind=real64),	intent(in)		::	x	The upper limit of the integration.
integer(kind=int32),	intent(in),	optional	::	max_iterations	Iteration limit; nonconvergence returns NaN.

Return Value real(kind=real64)

The value of the regularized beta function.

public pure elemental function regularized_gamma_lower(a, x, max_iterations) result(rst)

Computes the regularized lower incomplete gamma function.

Arguments

Type	Intent	Optional		Name
real(kind=real64),	intent(in)		::	a	The coefficient value. The value must be positive.
real(kind=real64),	intent(in)		::	x	The value at which to evaluate the function. The value must be non-negative.
integer(kind=int32),	intent(in),	optional	::	max_iterations	Iteration limit; nonconvergence returns NaN.

Return Value real(kind=real64)

The function value, which lies on the interval [0, 1].

public pure elemental function regularized_gamma_upper(a, x, max_iterations) result(rst)

Computes the regularized upper incomplete gamma function.

Arguments

Type	Intent	Optional		Name
real(kind=real64),	intent(in)		::	a	The coefficient value. The value must be positive.
real(kind=real64),	intent(in)		::	x	The value at which to evaluate the function. The value must be non-negative.
integer(kind=int32),	intent(in),	optional	::	max_iterations	Iteration limit; nonconvergence returns NaN.

Return Value real(kind=real64)

The function value, which lies on the interval [0, 1].

public pure elemental function stable_expm1(x) result(rst)

Computes exp(x) - 1 without subtractive cancellation near zero.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in)			::	x	Exponent argument.

Return Value real(kind=real64)

Difference; overflow gives +infinity and NaN inputs propagate.

public pure elemental function stable_log1p(x) result(rst)

Computes log(1 + x) accurately when x is near zero.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in)			::	x	Argument on [-1, infinity); NaN inputs propagate.

Return Value real(kind=real64)

Logarithm; x = -1 gives -infinity, x < -1 gives NaN.

fstats_special_functions Module

Contents

Functions

Uses

Functions

public pure elemental function beta(a, b) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function digamma(x) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function incomplete_beta(a, b, x) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function incomplete_gamma_lower(a, x) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function incomplete_gamma_upper(a, x) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function log_regularized_beta(a, b, x, max_iterations) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function log_regularized_gamma_lower(a, x, max_iterations) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function log_regularized_gamma_upper(a, x, max_iterations) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function regularized_beta(a, b, x, max_iterations) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function regularized_gamma_lower(a, x, max_iterations) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function regularized_gamma_upper(a, x, max_iterations) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function stable_expm1(x) result(rst)

Arguments

Return Value real(kind=real64)

public pure elemental function stable_log1p(x) result(rst)

Arguments

Return Value real(kind=real64)