pylops.optimization.leastsquares.normal_equations_inversion(Op, y, Regs, x0=None, Weight=None, dataregs=None, epsI=0.0, epsRs=None, NRegs=None, epsNRs=None, engine='scipy', show=False, **kwargs_solver)[source]#

Inversion of normal equations.

Solve the regularized normal equations for a system of equations given the operator Op, a data weighting operator Weight and optionally a list of regularization terms Regs and/or NRegs.


Operator to invert of size \([N \times M]\)


Data of size \([N \times 1]\)


Regularization operators (None to avoid adding regularization)

x0numpy.ndarray, optional

Initial guess of size \([M \times 1]\)

Weightpylops.LinearOperator, optional

Weight operator

dataregslist, optional

Regularization data (must have the same number of elements as Regs)

epsIfloat, optional

Tikhonov damping

epsRslist, optional

Regularization dampings (must have the same number of elements as Regs)


Normal regularization operators (None to avoid adding regularization). Such operators must apply the chain of the forward and the adjoint in one go. This can be convenient in cases where a faster implementation is available compared to applying the forward followed by the adjoint.

epsNRslist, optional

Regularization dampings for normal operators (must have the same number of elements as NRegs)

enginestr, optional

Solver to use (scipy or pylops)

showbool, optional

Display normal equations solver log


Arbitrary keyword arguments for chosen solver ( and are used for engine scipy and pylops, respectively)


When user does not supply atol, it is set to “legacy”.


Inverted model.


Convergence information (only when using

0: successful exit

>0: convergence to tolerance not achieved, number of iterations

<0: illegal input or breakdown

See also


Regularized inversion


Preconditioned inversion


See pylops.optimization.cls_leastsquares.NormalEquationsInversion

Examples using pylops.optimization.leastsquares.normal_equations_inversion#

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06. 2D Interpolation

06. 2D Interpolation