# Sparse Linear Systems¶

The solvers in this module focus on traditional least square problems for square or overdetermined linear systems $$A x = b$$ where the matrix $$A$$ is sparse and is represented by a linear operator abstraction providing the matrix multiplication and adjoint multiplication functions.

## Solvers¶

 lsqr(A, b, x0[, damp, atol, btol, conlim, …]) Solves the overdetermined system $$A x = b$$ in least square sense using LSQR algorithm. lsqr_jit(A, b, x0[, damp, atol, btol, …]) Solves the overdetermined system $$A x = b$$ in least square sense using LSQR algorithm. power_iterations(operator, b[, max_iters, …]) Computes the largest eigen value of a (symmetric) linear operator by power method power_iterations_jit(operator, b[, …]) Computes the largest eigen value of a (symmetric) linear operator by power method ista(operator, b, x0, step_size[, …]) Solves the problem $$\widehat{x} = \text{arg} \min_{x} \frac{1}{2}\| b - A x \|_2^2 + \lambda \mathbf{R}(x)$$ via iterative shrinkage and thresholding. ista_jit(operator, b, x0, step_size[, …]) Solves the problem $$\widehat{x} = \text{arg} \min_{x} \frac{1}{2}\| b - A x \|_2^2 + \lambda \mathbf{R}(x)$$ via iterative shrinkage and thresholding. fista(operator, b, x0, step_size[, …]) Solves the problem $$\widehat{x} = \text{arg} \min_{x} \frac{1}{2}\| b - A x \|_2^2 + \lambda \mathbf{R}(x)$$ via fast iterative shrinkage and thresholding. fista_jit(operator, b, x0, step_size[, …]) Solves the problem $$\widehat{x} = \text{arg} \min_{x} \frac{1}{2}\| b - A x \|_2^2 + \lambda \mathbf{R}(x)$$ via fast iterative shrinkage and thresholding.

## Data types¶

 LSQRSolution Solution for LSQR algorithm PowerIterSolution Solution of the eigen vector estimate ISTAState ISTA algorithm state FISTAState ISTA algorithm state