stdlib_specialfunctions#

Gaussian - Gaussian function#

Status#

Experimental

Description#

Computes the gaussian function:

\[f(x)=\exp(-x^2)\]

Syntax#

result = gaussian (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Example#

program example_gaussian
    use stdlib_kinds, only: sp
    use stdlib_math, only: linspace
    use stdlib_specialfunctions, only: gaussian
    implicit none
    integer, parameter :: n = 10
    real(sp) :: x(n), y(n)
    
    x = linspace(-2._sp, 2._sp, n)
    y = gaussian( x )
    print *, y
end program example_gaussian

Gaussian_grad - Gradient of the Gaussian function#

Status#

Experimental

Description#

Computes the gradient of the gaussian function:

\[f(x)=-2 * x * \exp( - x ^ 2)\]

Syntax#

result = gaussian_grad (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Elu - Exponential Linear Unit function#

Status#

Experimental

Description#

Computes the gaussian function:

\[\begin{split} \text{f}(x) = \begin{cases} x, & \text{if } x \geq 0 \\ a * (\exp(x) - 1), & \text{otherwise} \end{cases} \end{split}\]

Syntax#

result = elu (x,a)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind. a: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Example#

program example_elu
    use stdlib_kinds, only: sp
    use stdlib_math, only: linspace
    use stdlib_specialfunctions, only: elu
    implicit none
    integer, parameter :: n = 10
    real(sp) :: x(n), y(n)
  
    x = linspace(-2._sp, 2._sp, n)
    y = elu( x , 1.0 )
    print *, y
end program example_elu

Elu_grad - Gradient of the Exponential Linear Unit function#

Status#

Experimental

Description#

Computes the gradient of the gaussian function:

\[\begin{split} \text{f}(x) = \begin{cases} 1, & \text{if } x \geq 0 \\ a * \exp(x), & \text{otherwise} \end{cases} \end{split}\]

Syntax#

result = elu_grad (x,a)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind. a: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Relu - Rectified Linear Unit function#

Status#

Experimental

Description#

Computes the Rectified Linear Unit function:

\[f(x) = \text{max}(0,x)\]

Syntax#

result = relu (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Example#

program example_relu
    use stdlib_kinds, only: sp
    use stdlib_math, only: linspace
    use stdlib_specialfunctions, only: relu
    implicit none
    integer, parameter :: n = 10
    real(sp) :: x(n), y(n)
    
    x = linspace(-2._sp, 2._sp, n)
    y = relu( x )
    print *, y
end program example_relu

Relu_grad - Gradient of the Rectified Linear Unit function#

Status#

Experimental

Description#

Computes the gradient of the gaussian function:

\[\begin{split} f(x) = \begin{cases} 1, & \text{if } x \geq 0 \\ 0, & \text{otherwise} \end{cases} \end{split}\]

Syntax#

result = relu_grad (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

leaky_relu - leaky Rectified Linear Unit function#

Status#

Experimental

Description#

Computes the gaussian function:

\[\begin{split} \text{f}(x) = \begin{cases} x, & \text{if } x \geq 0 \\ a * x, & \text{otherwise} \end{cases} \end{split}\]

Syntax#

result = leaky_relu (x,a)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind. a: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Example#

program example_gelu
    use stdlib_kinds, only: sp
    use stdlib_math, only: linspace
    use stdlib_specialfunctions, only: leaky_relu
    implicit none
    integer, parameter :: n = 10
    real(sp) :: x(n), y(n)
    
    x = linspace(-2._sp, 2._sp, n)
    y = leaky_relu( x , 0.1_sp )
    print *, y
end program example_gelu

leaky_relu_grad - Gradient of the leaky Rectified Linear Unit function#

Status#

Experimental

Description#

Computes the gradient of the leaky_relu function:

\[\begin{split} \text{f}(x) = \begin{cases} 1, & \text{if } x \geq 0 \\ a , & \text{otherwise} \end{cases} \end{split}\]

Syntax#

result = leaky_relu_grad (x,a)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind. a: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as the input argument.

Gelu - Gaussian Error Linear Unit function#

Status#

Experimental

Description#

Computes the Gaussian Error Linear Unit function:

\[f(x) = \frac{1}{2} x ( 1 + \text{erf}(\frac{x}{\sqrt{2}}) ) \]

Syntax#

result = gelu (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Example#

program example_gelu
    use stdlib_kinds, only: sp
    use stdlib_math, only: linspace
    use stdlib_specialfunctions, only: gelu
    implicit none
    integer, parameter :: n = 10
    real(sp) :: x(n), y(n)
    
    x = linspace(-2._sp, 2._sp, n)
    y = gelu( x )
    print *, y
end program example_gelu

Gelu_grad - Gradient of the Gaussian Error Linear Unit function#

Status#

Experimental

Description#

Computes the gradient of the gaussian error linear unit function:

\[ f(x) = \frac{1}{2} ( 1 + \text{erf}(x \sqrt{2}) ) + x \sqrt{2} \exp( -\frac{1}{2} x^2) \]

Syntax#

result = gelu_grad (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Gelu_approx - Approximation of the Gaussian Error Linear Unit function#

Status#

Experimental

Description#

Computes a fast approximation of the Gaussian Error Linear Unit function using a fast \(\text{erf}\) approximation:

\[f(x) = \frac{1}{2} x ( 1 + \text{ferf}(\frac{x}{\sqrt{2}}) ) \]

Syntax#

result = gelu_approx (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Gelu_approx_grad - Gradient of the Approximated Gaussian Error Linear Unit function#

Status#

Experimental

Description#

Computes the gradient of the gaussian error linear unit function using a fast \(\text{erf}\) approximation:

\[ f(x) = \frac{1}{2} ( 1 + \text{ferf}(x \sqrt{2}) ) + x \sqrt{2} \exp( -\frac{1}{2} x^2) \]

Syntax#

result = gelu_approx_grad (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Selu - Scaled Exponential Linear Unit function#

Status#

Experimental

Description#

Applies the Scaled Exponential Linear Unit activation function:

\[\begin{split} f(x) = \begin{cases} scale * x, & \text{if } x \ge 0 \\ scale * (\alpha * exp(x) - \alpha ), & \text{otherwise} \end{cases} \end{split}\]
Where,
\[scale = 1.0507009873554804934193349852946\]
and
\[\alpha = 1.6732632423543772848170429916717\]

Syntax#

result = selu (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Example#

program example_selu
    use stdlib_kinds, only: sp
    use stdlib_math, only: linspace
    use stdlib_specialfunctions, only: selu
    implicit none
    integer, parameter :: n = 10
    real(sp) :: x(n), y(n)
    
    x = linspace(-2._sp, 2._sp, n)
    y = selu( x )
    print *, y
end program example_selu

selu_grad - Gradient of the Scaled Exponential Linear Unit function#

Status#

Experimental

Description#

Applies the gradient of the Scaled Exponential Linear Unit activation function:

\[\begin{split} f(x) = \begin{cases} scale, & \text{if } x \ge 0 \\ scale * \alpha * exp(x) , & \text{otherwise} \end{cases} \end{split}\]

Syntax#

result = selu_grad (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Sigmoid - Sigmoid function#

Status#

Experimental

Description#

Computes the sigmoid function:

\[f(x) = \frac{1}{1+\exp(-x)} \]

Syntax#

result = sigmoid (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Sigmoid_grad - Gradient of the Sigmoid function#

Status#

Experimental

Description#

Computes the gradient of the Sigmoid function:

\[f(x) = \frac{\exp(x)}{(1+\exp(x))^2} \]

Syntax#

result = sigmoid_grad (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

SiLU - Sigmoid Linear Unit function#

Status#

Experimental

Description#

Computes the Sigmoid Linear Unit function:

\[f(x) = \frac{x}{1+\exp(-x)} \]

Syntax#

result = silu (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Example#

program example_silu
    use stdlib_kinds, only: sp
    use stdlib_math, only: linspace
    use stdlib_specialfunctions, only: silu
    implicit none
    integer, parameter :: n = 10
    real(sp) :: x(n), y(n)
    
    x = linspace(-2._sp, 2._sp, n)
    y = silu( x )
    print *, y
end program example_silu

Silu_grad - Gradient of the Sigmoid Linear Unit function#

Status#

Experimental

Description#

Computes the gradient of the Sigmoid function:

\[f(x) = \frac{\exp(x)*(x+(1+\exp(x))^2)}{(1+\exp(x))^2} \]

Syntax#

result = silu_grad (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Step - Step function#

Status#

Experimental

Description#

Computes the step function:

\[\begin{split} f(x) = \begin{cases} 1, & \text{if } x > 0 \\ 0, & \text{otherwise} \end{cases} \end{split}\]

Syntax#

result = step (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Example#

program example_step
    use stdlib_kinds, only: sp
    use stdlib_math, only: linspace
    use stdlib_specialfunctions, only: step
    implicit none
    integer, parameter :: n = 10
    real(sp) :: x(n), y(n)
    
    x = linspace(-2._sp, 2._sp, n)
    y = step( x )
    print *, y
end program example_step

step_grad - Gradient of the Step function#

Status#

Experimental

Description#

Computes the gradient of the Sigmoid function:

\[f(x) = 0 \]

Syntax#

result = step_grad (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

softmax - softmax function#

Status#

Experimental

Description#

Computes the softmax function:

\[f(x) = \frac{\exp(x)-\text{max}(x_j)}{\sum_j{\exp(x)-\text{max}(x_j)}}\]

Syntax#

result = softmax (x,dim)

Class#

Pure function for ranks 1 to 4.

Arguments#

x: Shall be an array of rank 1 to 4 of any real kind. dim: integer scalar indicating upon which dimension to apply the normalization.

Return value#

The function returns an array with the same rank and kind as the input argument x.

Example#

program example_softmax
    use stdlib_kinds, only: sp
    use stdlib_math, only: linspace
    use stdlib_specialfunctions, only: softmax
    implicit none
    integer, parameter :: n = 10
    real(sp) :: x(n), y(n)
    
    x = linspace(-2._sp, 2._sp, n)
    y = softmax( x )
    print *, y
end program example_softmax

softmax_grad - Gradient of the softmax function#

Status#

Experimental

Description#

Computes the gradient of the softmax function:

\[f(x,dim) = \text{softmax}(x,dim)*(1-\text{softmax}(x,dim)) \]

Syntax#

result = softmax_grad (x,dim)

Class#

Pure function for ranks 1 to 4.

Arguments#

x: Shall be an array of rank 1 to 4 of any real kind. dim: integer scalar indicating upon which dimension to apply the normalization.

Return value#

The function returns a value with the same type and kind as input argument.

softplus - softplus function#

Status#

Experimental

Description#

Computes the softplus function:

\[f(x) = \log(\exp(x)+1)\]

Syntax#

result = softplus (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Example#

program example_softplus
    use stdlib_kinds, only: sp
    use stdlib_math, only: linspace
    use stdlib_specialfunctions, only: softplus
    implicit none
    integer, parameter :: n = 10
    real(sp) :: x(n), y(n)
    
    x = linspace(-2._sp, 2._sp, n)
    y = softplus( x )
    print *, y
end program example_softplus

softplus_grad - Gradient of the softplus function#

Status#

Experimental

Description#

Computes the gradient of the softplus function:

\[f(x) = \frac{\exp(x)}{\exp(x)+1} \]

Syntax#

result = softplus_grad (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Fast tanh - Approximation of the hyperbolic tangent function#

Status#

Experimental

Description#

Computes an approximated but faster solution to:

\[f(x)=\tanh(x)\]

Syntax#

result = fast_tanh (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

fast_tanh_grad - Gradient of the approximation of the hyperbolic tangent function#

Status#

Experimental

Description#

Computes the gradient of the fast_tanh function:

\[f(x)=1 - \text{fast\_tanh}(x)^2\]

Syntax#

result = fast_tanh_grad (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

Fast erf - Approximation of the error function#

Status#

Experimental

Description#

Computes an approximated but faster solution to:

\[f(x)=\text{erf}(x)\]

Syntax#

result = fast_erf (x)

Class#

Elemental function

Arguments#

x: Shall be a scalar or array of any real kind.

Return value#

The function returns a value with the same type and kind as input argument.

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