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# Numerical Methods

**Numerical methods** in scqubits compute eigenvalues and eigenvectors, matrix elements, and other properties. Main methods:

**Diagonalization**: construct the Hamiltonian matrix, compute eigendecomposition via LAPACK/scipy. Works for systems up to ~1000×1000.

**Sparse diagonalization**: for larger matrices, use sparse solvers (Arnoldi, Lanczos) to find only lowest eigenvalues.

**Time evolution**: solve the Schrödinger equation numerically (RK45, etc.) to compute transient behavior.

```python
from scqubits import Transmon
import numpy as np

transmon = Transmon(EJ=15.0, EC=0.3, ncut=30)

# Eigenvalues via numpy/scipy
evals = transmon.eigenvals(n=10)

# Matrix elements
n_op = transmon.n_operator()
matrix_elem = (transmon.eigenvecs(n=2)[0].dag() * n_op * transmon.eigenvecs(n=2)[1]).full()[0, 0]

# Time evolution (for driven systems)
H_drive = lambda t, args: 0.1 * transmon.n_operator()  # Driven Hamiltonian
times = np.linspace(0, 10, 100)
# (Would use scqubits.ParameterSweep or integrate manually)
```

scqubits uses scipy, numpy, and numpy.linalg for standard operations. For custom problems, extract the Hamiltonian matrix and use any numerical solver.

Numerical accuracy is governed by floating-point precision and convergence of ODE solvers. Check convergence by varying tolerances or step sizes.