This manual is the definitive guide to GNU Octave, an interactive environment for numerical computation. GNU Octave provides a convenient command-line interface for solving linear and nonlinear problems using vectors and matrices. This updated edition of the manual covers version 4.0.0 of GNU Octave, and includes documentation for new features such as the new graphical userinterface, sparse matrices, linear programming and computational geometry. GNU Octave is free software, distributed under the GNU General Public License (GPL). As GNU Octave became such a big project over the years, we had to split this reference manual in two parts that are two separate physical books. To keep it consistent with our digital manual, the references and page numbers cover both physical books as it were one. Therefore please note that you probably want to have both parts.