pylops.SecondDerivative¶

class pylops.SecondDerivative(dims, axis=-1, sampling=1.0, kind='centered', edge=False, dtype='float64', name='S')[source]¶

Second derivative.

Apply a second derivative using a three-point stencil finite-difference approximation along axis.

Parameters:

dimslist or int: Number of samples for each dimension (None if only one dimension is available)
axisint, optional: Added in version 2.0.0.

Axis along which derivative is applied.
samplingfloat, optional: Sampling step \(\Delta x\).
kindstr, optional: Derivative kind (forward, centered, or backward).
edgebool, optional: Use shifted derivatives at edges (True) or ignore them (False). This is currently only available for centered derivative
dtypestr, optional: Type of elements in input array.
namestr, optional: Added in version 2.0.0.

Name of operator (to be used by pylops.utils.describe.describe)

Attributes:

dimstuple

Shape of the array after the adjoint, but before flattening.

For example, x_reshaped = (Op.H * y.ravel()).reshape(Op.dims).

dimsdtuple

Shape of the array after the forward, but before flattening. In this case, same as dims.

shapetuple

Operator shape.

Notes

The SecondDerivative operator applies a second derivative to any chosen direction of a multi-dimensional array.

For simplicity, given a one dimensional array, the second-order centered second derivative is:

\[y[i] = (x[i+1] - 2x[i] + x[i-1]) / \Delta x^2\]

while the second-order forward stencil is:

\[y[i] = (x[i+2] - 2x[i+1] + x[i]) / \Delta x^2\]

and the second-order backward stencil is:

\[y[i] = (x[i] - 2x[i-1] + x[i-2]) / \Delta x^2\]

Methods

`__init__`(dims[, axis, sampling, kind, edge, ...])
`adjoint`()
`apply_columns`(cols)	Apply subset of columns of operator
`cond`([uselobpcg])	Condition number of linear operator.
`conj`()	Complex conjugate operator
`div`(y[, niter, densesolver])	Solve the linear problem \(\mathbf{y}=\mathbf{A}\mathbf{x}\).
`dot`(x)	Matrix-matrix or matrix-vector multiplication.
`eigs`([neigs, symmetric, niter, uselobpcg])	Most significant eigenvalues of linear operator.
`matmat`(X)	Matrix-matrix multiplication.
`matvec`(x)	Matrix-vector multiplication.
`reset_count`()	Reset counters
`rmatmat`(X)	Matrix-matrix multiplication.
`rmatvec`(x)	Adjoint matrix-vector multiplication.
`todense`([backend])	Return dense matrix.
`toimag`([forw, adj])	Imag operator
`toreal`([forw, adj])	Real operator
`tosparse`()	Return sparse matrix.
`trace`([neval, method, backend])	Trace of linear operator.
`transpose`()