Linear methods in band theory, as presented by Andersen and Krogh, offer efficient and physically transparent approaches to solving the band-structure problem. These methods utilize the variational principle for the one-electron Hamiltonian, with trial functions being linear combinations of energy-independent augmented plane waves (APW) and muffin-tin orbitals (MTO). The secular equations derived from these methods are eigenvalue equations linear in energy, allowing for the simultaneous determination of energy eigenvalues and eigenvectors through diagonalization. The linear-APW (LAPW) method is particularly effective for treating d bands and includes non-MT contributions to the potential through their Fourier components. The linear-MTO (LMTO) method is well-suited for closely packed structures and combines features of Korringa-Kohn-Rostoker, linear-combination-of-atomic-orbitals, and cellular methods. The secular matrix is linear in energy, and the overlap integrals factorize into potential parameters and structure constants, which are canonical and independent of energy or cell volume. The LAPW method forms a complete set of functions in the interstitial region and is particularly suited for open structures where non-MT contributions are significant. The LMTO method, on the other hand, uses solutions of the Laplace equation in the interstitial region, which have zero kinetic energy. The canonical structure constants of the LMTO method are independent of energy and atomic volume, making them suitable for tabulation. Both methods are computationally efficient, with the LAPW and LMTO methods having similar convergence properties to conventional nonlinear APW and KKR methods. The computational speed is determined by diagonalization techniques, with the LMTO method being preferred due to its smaller matrix size. The methods are applied to free-electron energy bands and have been shown to accurately reproduce these bands. The accuracy of the LAPW and LMTO methods is demonstrated through their ability to handle energy-dependent parameters and their application to transition metals and other materials. The methods are also extended to include relativistic effects, with the spin-orbit coupling parameter and normalization parameter playing a crucial role in the relativistic treatment. The paper discusses the accuracy and convergence of these methods, showing that the LAPW method yields energy bands correct to third order in energy, while the LMTO method provides accurate results for closely packed structures. The methods are applied to various materials, including transition metals and compounds, with the potential parameters and structure constants being essential for their performance. The results highlight the effectiveness of these linear methods in solving the band-structure problem with high accuracy and efficiency.Linear methods in band theory, as presented by Andersen and Krogh, offer efficient and physically transparent approaches to solving the band-structure problem. These methods utilize the variational principle for the one-electron Hamiltonian, with trial functions being linear combinations of energy-independent augmented plane waves (APW) and muffin-tin orbitals (MTO). The secular equations derived from these methods are eigenvalue equations linear in energy, allowing for the simultaneous determination of energy eigenvalues and eigenvectors through diagonalization. The linear-APW (LAPW) method is particularly effective for treating d bands and includes non-MT contributions to the potential through their Fourier components. The linear-MTO (LMTO) method is well-suited for closely packed structures and combines features of Korringa-Kohn-Rostoker, linear-combination-of-atomic-orbitals, and cellular methods. The secular matrix is linear in energy, and the overlap integrals factorize into potential parameters and structure constants, which are canonical and independent of energy or cell volume. The LAPW method forms a complete set of functions in the interstitial region and is particularly suited for open structures where non-MT contributions are significant. The LMTO method, on the other hand, uses solutions of the Laplace equation in the interstitial region, which have zero kinetic energy. The canonical structure constants of the LMTO method are independent of energy and atomic volume, making them suitable for tabulation. Both methods are computationally efficient, with the LAPW and LMTO methods having similar convergence properties to conventional nonlinear APW and KKR methods. The computational speed is determined by diagonalization techniques, with the LMTO method being preferred due to its smaller matrix size. The methods are applied to free-electron energy bands and have been shown to accurately reproduce these bands. The accuracy of the LAPW and LMTO methods is demonstrated through their ability to handle energy-dependent parameters and their application to transition metals and other materials. The methods are also extended to include relativistic effects, with the spin-orbit coupling parameter and normalization parameter playing a crucial role in the relativistic treatment. The paper discusses the accuracy and convergence of these methods, showing that the LAPW method yields energy bands correct to third order in energy, while the LMTO method provides accurate results for closely packed structures. The methods are applied to various materials, including transition metals and compounds, with the potential parameters and structure constants being essential for their performance. The results highlight the effectiveness of these linear methods in solving the band-structure problem with high accuracy and efficiency.