Several numerical methods used in the study of tensor network renormalization are introduced, including the power, Lanczos, conjugate gradient, Arnoldi methods, and quantum Monte Carlo simulation.

Chapter 4 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. An extensive section discusses the conditioning of linear-algebra problems: borrowing ideas and examples from matrix perturbation theory, this teaches students what to look out for. Roughly half of the remaining chapter is dedicated to the solution of linear systems of equations, employing methods of varying sophistication: Gaussian elimination, LU decomposiion, pivoting, and the Jacobi iterative method. The second half addresses the eigenvalue problem, again with a variety of methods, including the power method, eigenvalue shifting, and the QR method. Crucially, this includes explicit mathematical derivations of these approaches. A brief introduction to the singular value decomposition is also given, including both an existence proof and a programming implementation. The chapter is rounded out by an extensive physics project, which studies the eigenvalue problem of interacting spins, and a problem set. The physics project patiently builds up matrix quantum mechanics, allowing students to tackle problems of increasing difficulty.

Chapter 19: In this chapter, we introduce new examples of norms, with special attention to submultiplicative norms on matrices. These norms are well-adapted to applications involving power series of matrices and iterative numerical algorithms. We use them to prove a formula for the spectral radius that is the key to a fundamental theorem on positive matrices in the next chapter.

Chapter 20: This chapter is about some remarkable properties of positive matrices, by which we mean square matrices with real positive entries. Positive matrices are found in economic models, genetics, biology, team rankings, network analysis, Google's PageRank, and city planning. The spectral radius of any matrix is the absolute value of an eigenvalue, but for a positive matrix the spectral radius itself is an eigenvalue, and it is positive and dominant. It is associated with a positive eigenvector, whose ordered entries have been used for ranking sports teams, priority setting, and resource allocation in multicriteria decision-making. Since the spectral radius is a dominant eigenvalue, an associated positive eigenvector can be computed by the power method. Some properties of positive matrices are shared by nonnegative matrices that satisfy certain auxiliary conditions. One condition that we investigate in this chapter is that some positive power has no zero entries.

This chapter can be used as a six week “lab" component to a mathematical methods course, one section each week.The chapter is relatively self-contained, and consists of numerical methods that complement the analytic solutions found in the rest of the book.There are methods for solving ODE problems (both in initial and boundary value form) approximating integrals, and finding roots.There is also a discussion of the eigenvalue problem in the context of approximate solutions in quantum mechanics and a section on the discrete Fourier transform.