Claculus Numericus

Calculus Numericus is a comprehensive modern treatise on the art and science of computing with numbers: the discipline by which the continuous mathematics of the world is entrusted to the discrete arithmetic of the machine, and by which we learn how far that trust may safely extend. Across eight parts and forty-one chapters, the book develops the subject from first principles to its present frontiers. It begins with finite-precision arithmetic, error, conditioning and stability; proceeds through interpolation, differentiation, integration and numerical linear algebra; derives the methods for ordinary and partial differential equations; and then turns to optimisation, inverse problems, Monte Carlo simulation, uncertainty quantification, automatic differentiation, parallel computation and scientific machine learning. Three integrated studios apply the whole apparatus to physical systems, living systems, medical imaging, earth science, finance and decision under uncertainty. Every formula is derived, every algorithm motivated, and every figure generated from the methods the text describes. Rigorous without being oracular, and practical without becoming a catalogue of recipes, the volume is at once a graduate course, a working reference and an extended argument for numerical analysis as a science of judgement