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A free scan shows the junk files, broken settings and background clutter dragging Windows down - then fixes them in one click.Free scan · Windows 10 & 11scipy.optimize.leastsq finds parameter values that minimize the sum of squared residuals returned by your function. To use it, provide a residual vector, an initial parameter estimate, and—optionally—a Jacobian. It is a local iterative solver: the starting estimate and model formulation matter, and a returned parameter vector is not automatically a successful or trustworthy fit.
What `leastsq` minimizes
For a parameter vector with N unknowns, your function returns M residuals. SciPy seeks parameters that minimize the sum of the residuals squared. The residual count must be at least the number of unknowns (M ≥ N).
For fitting observations, a residual is commonly the difference between an observed value and the model’s predicted value at that observation. Return the whole vector of differences—not a scalar sum of squares. `leastsq` performs the squaring and summation. Residuals should be floating-point values and must not contain NaNs.
The current SciPy v1.18.0 reference describes `leastsq` as a wrapper around MINPACK’s `lmdif` and `lmder` algorithms. It starts from the initial estimate you provide and performs a local search; it does not promise to find a global minimum. See the SciPy v1.18.0 `leastsq` reference.
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Fit a model by returning its residual vector
For example, suppose a model predicts values from input data and a parameter vector. The residual function receives the parameters first, then any fixed data supplied through `args`:
import numpy as np
from scipy.optimize import leastsq
def residuals(params, xdata, ydata):
amplitude, frequency, phase, offset = params
predicted = amplitude * np.sin(frequency * xdata + phase) + offset
return ydata - predicted
# xdata and ydata are NumPy arrays of equal length.
x0 = np.array([1.0, 1.0, 0.0, 0.0])
result = leastsq(residuals, x0, args=(xdata, ydata))
params = result[0]
status = result[1]Replace `xdata` and `ydata` with your observation arrays. This sinusoidal form is also used in SciPy’s optimization tutorial; the important API pattern is that the function returns one residual per observation. See the SciPy v0.17.0 optimization tutorial.
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The result’s parameter vector is always one-dimensional, even if `x0` was supplied in another shape. Choose an initial estimate that is plausible for your model and data. Since the method is iterative and local, different starting values can lead to different outcomes.
Check whether the solver found a solution
By default, the function returns the solution vector and an integer termination flag. A termination flag alone is not an accuracy guarantee. For the documented SciPy API, status values 1, 2, 3, and 4 indicate that a solution was found. Use `full_output=True` to receive additional information:
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x, cov_x, infodict, mesg, ier = leastsq(
residuals,
x0,
args=(xdata, ydata),
full_output=True,
)
if ier in (1, 2, 3, 4):
print("Termination indicates a solution:", mesg)
else:
print("Solver did not report success:", mesg)
print("Last iterate:", x)Inspect `mesg` and the diagnostics in `infodict`, particularly when the status indicates failure. In that case, the returned `x` is the last iterate, not a successful solution.
Provide a Jacobian when you can
You may pass a function for the Jacobian—the derivatives of the residuals with respect to the parameters—using `Dfun`. If you omit it, SciPy estimates derivatives numerically. A correct analytic Jacobian can avoid that estimation, but its layout must match the `col_deriv` setting: by default, derivatives are supplied across rows; set `col_deriv=True` when they are supplied down columns.
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Check the Jacobian’s dimensions and derivative orientation against the residual vector and parameter vector. An incorrect Jacobian can undermine the optimization rather than improve it.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Understand the stopping and scaling options
The API’s tolerances determine when iteration stops; they do not certify that parameters are accurate or that the fit is good.
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| Option | Role | Practical interpretation |
|---|---|---|
ftol |
Objective-change stopping criterion | Controls termination based on changes in the sum-of-squares objective. |
xtol |
Solution-change stopping criterion | Controls termination based on changes in the parameter vector. |
gtol |
Residual/Jacobian orthogonality criterion | Controls termination based on how residuals relate to the Jacobian. |
maxfev |
Maximum function evaluations | Its documented default is 200*(N+1) without Dfun and 100*(N+1) with Dfun, where N is the number of parameters. |
diag |
Variable scaling factors | Supply positive scale factors when parameters have substantially different scales. |
factor |
Initial step bound control | The documented acceptable range is (0.1, 100). |
These meanings and documented defaults are for SciPy v1.18.0; check the reference for the version installed in your environment because API behavior and defaults can change. The v1.18.0 API reference documents the full parameter list.
Interpret `cov_x` cautiously
When `full_output=True`, `cov_x` is an inverse-Hessian/Jacobian-based approximation. It is not, by itself, the parameter covariance matrix: SciPy says to multiply it by the residual variance to obtain a covariance estimate. If `cov_x` is `None`, the approximation is unavailable because the curvature is numerically flat in at least one parameter direction.
Treat this as an approximation tied to the least-squares residual model, not a general guarantee about parameter uncertainty.
Choose the SciPy fitting API that matches the problem
| Need | API | Why it fits |
|---|---|---|
| Unbounded least squares through the MINPACK interface | leastsq |
A focused interface around MINPACK’s lmdif and lmder algorithms. |
| Parameter bounds or robust loss functions | least_squares |
Supports bounds and selectable methods and loss functions; its lm method is also MINPACK-based. |
| Fitting a named model function to observed data | curve_fit |
Provides a higher-level interface with data arrays, initial parameter guesses, bounds, and method selection. |
SciPy documents these distinctions in its `least_squares` reference and `curve_fit` reference. `curve_fit` uses `leastsq` for its `lm` method and `least_squares` otherwise.
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