MATHEMATICS FOR ENGINEERING AND SCIENCE VOL. 2- LINEAR AND NON-LINEAR PROGRAMMING EMANUEL VAZ; UP, MIT; SORBONNE, JIC, NYAS

74 pages

Target Audience: This work adapts to students of numerical calculation of our universities and engineers in particular.
Contents: Direct linear interpolation. Direct linear interpolation formula.  Error of the method in linear interpolation. Definition.  Evaluation of the error of method. Sign of error from method. Uncertainty error for direct interpolation. Systematic error in direct linear interpolation. Definition. Evaluation of the Systematic error. Miscalculation.Definitions.  Objective function. Conditions. Conditions of the problem. Non-negative conditions.  Possible region. Turning point. Without –boundary region. Basic theorem of the linear programming. Basic theorem of the arithmetic.  Basic theorem of algebra. Basic principle of the counting. Basic theorem of the linear programming. Boundary region. What the optimization is. As  the linear programming appeared. What linear programming is? Limitation of the linear programming. Constant coefficient. Divisibility. Addictively.  Mathematical formularization. The standard form of the Simplex method.
Non-negativity Restrictions. Objective function. The method to Solve of Excel.  Presentation of the example. Formularization of the problem of linear programming. Graphical solution. Analysis of sensibility.  Variation of the coefficients of the objective function. Analysis of the second members of equations. Price shade. Attainment of the solution with the resource of excel. Resolution of a problem with four variable of decision with the method “To Solve of Excel”. Some problems of analysis of activities. Objective function. Restrictions. Compact form. Compact form to the restrictions. Parameters. Variables. Problem of the Diet. Problem of Transport. Graphical Technique to solve models of linear programming with two variables of decision.  Examples of reduction of a model of linear programming for the standard form.  Standard Form. Occurring the inequalities. Graphical solution. Graph for sensibility analyze. Least squares. Problem statement. Solving the least squares problem. Least squares and regression analysis. Estimating parameter. Summarizing the data. Estimating beta (the slope). Estimating alpha (The intercept). Limitations.  Weighted least squares. Linear Algebraic Derivation.  Weighted case. Nonlinear programming.  Introduction. The generic problem. A Multi-Variate Calculus Review.  Taylor’s Theorem. Mean value theorem. Implicit function theorem. Unconstrained problems.  Feasible direction. First order necessary conditions.  Second order sufficient conditions (unconstrained case). Convex programming. Theorem 1. Corollary. The Lagrange Multiplier theorem. Lemma 1. Lemma 2 (theorem of alternatives). The Lagrange Multiplier Theorem.  Theorem 2.  Lemma 3. Lemma4. Karush-Kuhn-Tucker theorem. Sufficient conditions. Problems. Non linear programming. Prob 4. Gasoline blending. Prob. 5. Site location. Prob.6. Production quantities. Prob.7. Flow in Pipes. Prob.8. Robot motion planning. Prob.9. Design parameters for coil spring. Issues with non-linear programs. Approaches to non-linear programs.  Polynomial interpolation and approximation. Lagrange interpolation formula.  Theorem 1. Theorem 2.  Linear rational interpolation in barycentric form . Lemma 1. Theorem 2. Appendix. Formula of the two levels. Resolution of a system of equations with a calculating machine. Bibliography.