klampt.math.autodiff.math_ad module

Basic math / linear algebra routines. Defines the following AD functions:

Function

Derivative

Notes

exp

1

log

1

sqrt

1

sin

Y

cos

Y

dot

Y

linear

Y

Produces \(A x\) for a fixed A

quadratric

Y

Produces \(x^T A x\) for a fixed A

bilinear

Y

Produces \(x^T A y\) for a fixed A

ShapedDot

Y

Reshapes a vector before performing dot

norm

1

Standard L2 norm

normSquared

Y

distance

1

Standard L2 norm

distanceSquared

Y

unit

N

cross

1

interpolate

1

Module contents

`exp`	Autodiff'ed function comparable to np.exp.
`log`	Autodiff'ed function comparable to np.log.
`sqrt`	Autodiff'ed function comparable to np.sqrt.
`sin`	Autodiff'ed function comparable to np.sin.
`cos`	Autodiff'ed function comparable to np.cos.
`dot`	Autodiff'ed function comparable to np.dot.
`linear`(A, x)	Autodiff'ed function comparable to np.dot(A,x).
`quadratric`(A, x)	Autodiff'ed function comparable to np.dot(x,np.dot(A,x)).
`bilinear`(x, A, y)	Autodiff'ed function comparable to np.dot(x,np.dot(A,y)).
`ShapedDot`([shape1, shape2])	This function is used to do matrix-vector products, which are by normally not supported by the dot function since all arguments are flattened 1D arrays.
`norm`	Autodiff'ed function comparable to np.linalg.norm.
`normSquared`	Autodiff'ed function comparable to np.dot(x,x).
`distance`	Autodiff'ed function comparable to np.linalg.norm(x-y).
`distanceSquared`	Autodiff'ed function comparable to np.dot(x-y,x-y).
`unit`	Autodiff'ed function comparable to x/norm(x).
`cross`	Autodiff'ed function comparable to np.cross(x,y).
`interpolate`	Autodiff'ed function comparable to (1-u)a+ub.

class klampt.math.autodiff.math_ad.ShapedDot(shape1=None, shape2=None)[source]

Bases: ADFunctionInterface

This function is used to do matrix-vector products, which are by normally not supported by the dot function since all arguments are flattened 1D arrays. This function reshapes either the first or second arguments into a 2D array before calling dot. You must specify the array shape upon instantiation.

n_args()[source]: Returns the number of arguments.

n_in(arg)[source]: Returns the number of entries in argument #arg. If 1, this can either be a 1-D vector or a scalar. If -1, the function can accept a variable sized argument.

n_out()[source]: Returns the number of entries in the output of the function. If -1, this can output a variable sized argument.

eval(x, y)[source]

Evaluates the application of the function to the given (instantiated) arguments.

Parameters:: args (list) – a list of arguments, which are either ndarrays or scalars.

derivative(arg, x, y)[source]

Returns the Jacobian of the function w.r.t. argument #arg.

Parameters:

arg (int) – A value from 0,…,self.n_args()-1 indicating that we wish to take \(df/dx_{arg}\).
args (list of ndarrays) – arguments for the function.

Returns:

A numpy array of shape (self.n_out(),self.n_in(arg)). Keep in mind that even if the argument or result is a scalar, this needs to be a 2D array.

If the derivative is not implemented, raise a NotImplementedError.

If the derivative is zero, can just return 0 (the integer) regardless of the size of the result.

Return type:

ndarray

jvp(arg, darg, x, y)[source]

Performs a Jacobian-vector product for argument #arg.

Parameters:

arg (int) – A value from 0,…,self.n_args()-1 indicating that we wish to calculate df/dx_arg * darg.
darg (ndarray) – the derivative of x_arg w.r.t some other parameter. Must have darg.shape = (self.n_in(arg),).
args (list of ndarrays) – arguments for the function.

Returns:

A numpy array of length self.n_out()

If the derivative is not implemented, raise a NotImplementedError.

Return type:

ndarray

gen_derivative(arg, x, y)[source]

Generalized derivative that allows higher-order derivatives to be taken.

Parameters:

arg (list) – Indicates the order of derivatives to be taken. For example, to take the 2nd derivative w.r.t. x0,x1, arg = [0,1] should be specified.
args (list of ndarrays) – arguments for the function.

Returns:

A tensor of shape (n_out,n_in(arg[0]),..., n_in(arg[-1]).

If the generalized derivative is not implemented, raise a NotImplementedError.

If the generalized derivative is zero, can just return 0 (the integer) regardless of the size of the result.

Return type:

ndarray

klampt.math.autodiff.math_ad.exp(*args) = <klampt.math.autodiff.math_ad._ADExp object>: Autodiff’ed function comparable to np.exp. First derivative is implemented.

klampt.math.autodiff.math_ad.log(*args) = <klampt.math.autodiff.math_ad._ADLog object>: Autodiff’ed function comparable to np.log. First derivative is implemented.

klampt.math.autodiff.math_ad.sqrt(*args) = <klampt.math.autodiff.math_ad._ADSqrt object>: Autodiff’ed function comparable to np.sqrt. First derivative is implemented.

klampt.math.autodiff.math_ad.sin(*args) = <klampt.math.autodiff.math_ad._ADSin object>: Autodiff’ed function comparable to np.sin. All derivatives are implemented.

klampt.math.autodiff.math_ad.cos(*args) = <klampt.math.autodiff.math_ad._ADCos object>: Autodiff’ed function comparable to np.cos. All derivatives are implemented.

klampt.math.autodiff.math_ad.dot(*args) = <klampt.math.autodiff.math_ad._ADDot object>: Autodiff’ed function comparable to np.dot. All derivatives are implemented.

klampt.math.autodiff.math_ad.linear(A, x)[source]: Autodiff’ed function comparable to np.dot(A,x). A must be a constant, 2D np.array. x may be an expression.

klampt.math.autodiff.math_ad.quadratric(A, x)[source]: Autodiff’ed function comparable to np.dot(x,np.dot(A,x)). A must be a constant, 2D np.array. x may be an expression.

klampt.math.autodiff.math_ad.bilinear(x, A, y)[source]: Autodiff’ed function comparable to np.dot(x,np.dot(A,y)). A must be a constant, 2D np.array. x and y may be expressions.

klampt.math.autodiff.math_ad.norm(*args) = <klampt.math.autodiff.ad._ad_norm object>: Autodiff’ed function comparable to np.linalg.norm. First derivative is implemented.

klampt.math.autodiff.math_ad.normSquared(*args) = <klampt.math.autodiff.ad._ad_normSquared object>: Autodiff’ed function comparable to np.dot(x,x). All derivatives are implemented.

klampt.math.autodiff.math_ad.distance(*args) = <klampt.math.autodiff.math_ad._ADDistanceL2 object>: Autodiff’ed function comparable to np.linalg.norm(x-y). First derivative is implemented.

klampt.math.autodiff.math_ad.distanceSquared(*args) = <klampt.math.autodiff.math_ad._ADDistanceSquared object>: Autodiff’ed function comparable to np.dot(x-y,x-y). All derivatives are implemented.

klampt.math.autodiff.math_ad.unit(*args) = <klampt.math.autodiff.ad._ad_unit object>: Autodiff’ed function comparable to x/norm(x). First derivative is implemented.

klampt.math.autodiff.math_ad.cross(*args) = <klampt.math.autodiff.ad._ad_cross object>: Autodiff’ed function comparable to np.cross(x,y). First derivative is implemented. Some 2nd derivatives are implemented.

klampt.math.autodiff.math_ad.interpolate(*args) = <klampt.math.autodiff.ad._ad_interpolate object>: Autodiff’ed function comparable to (1-u)*a+u*b. First derivatives is implemented.