1 Matrices: Definitions and Fundamental Operations.- Fundamental Matrix Operations.- Linear Algebra.- Eigenvalues.- Linear Transformations.- Rotation of Axes.- The Metrie Tensor.- 2 Symmetry of Finite Objects.- Groups.- Representations.- Point Groups.- Basis Functions.- 3 Symmetry of Infinitely Repeated Patterns.- Bravais Lattices.- Space Groups.- 4 Vectors.- Scalar and Vector Products.- The Reciprocal Lattice.- The Orientation Matrix.- Zones and Forms.- Sublattices and Superlattices.- 5 Tensors.- Covariance and Contravariance.- The Multivariate Normal Disribution.- Anisotropic Atomic Displacement Factors.- The Equivalent Isotropic Temperature Factor.- Effect of Symmetry.- Tensors of Higher Ranks.- Moments and Cumulants.- Rigid Body Motion.- 6 Data Fitting.- Fitting Functions.- Finding the Minimum.- False Minima.- 7 Estimation of Uncertainty.- Estimates.- The Precision of Estimates of Precision.- Models with More than One Parameter.- Estimates of Precision When the Model Is Not Least Squares.- 8 Significance and Accuracy.- The F Distribution.- Student's t Distribution.- Correlation.- Relationship Between Precision and Accuracy.- Uncertainties of Derived Functions.- The Projection Matrix.- 9 Constrained Crystal Structure Refinement.- The Observed Data.- The Model.- The General Form for a Constrained Model.- Shape Constraints.- Rigid Body Thermal Motion Constraints.- Chemical Constraints.- Representing non-Gaussian Distributions.- 10 The Fast Fourier Transform.- The Discrete Fourier Transform.- The Good-Thomas Algorithm.- The Cooley-Tukey Algorithm.- Prime Numbers.- FFTs for Real Sequences.- Space Group Symmetry.- Appendix A Stereographic Projection.- Appendix B Eigenvalues and Eigenvectors of 3 × 3 Symmetric Matrices.- Appendix C Sublattices and Superlattices.- Appendix D The Probability Integral, the Gamma Function, and Related Topics.- Appendix E The Harmonie Oscillator in Quantum Mechanics: Bloch's Theorem.- Appendix F Symmetry Restrictions on Seeond, Third, and Fourth Rank Tensors.- Appendix G Some Useful Computer Programs.
