class pylops.SecondDerivative(N, dims=None, dir=0, sampling=1, edge=False, dtype='float64', kind='centered')[source]

Second derivative.

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

N : int

Number of samples in model.

dims : tuple, optional

Number of samples for each dimension (None if only one dimension is available)

dir : int, optional

Direction along which the derivative is applied.

sampling : float, optional

Sampling step \(\Delta x\).

edge : bool, optional

Use reduced order derivative at edges (True) or ignore them (False) for centered derivative

dtype : str, optional

Type of elements in input array.

kind : str, optional

Derivative kind (forward, centered, or backward).


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 first 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:

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

Operator shape

explicit : bool

Operator contains a matrix that can be solved explicitly (True) or not (False)


__init__(N[, dims, dir, sampling, edge, …]) Initialize this LinearOperator.
adjoint() Hermitian 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.
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() Transpose this linear operator.