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Students
are expected to understand the fundamental concepts involved in numerical
analysis, and learn some basic approaches and techniques in solving linear
and nonlinear systems, constructing interpolations and do numerical
differntiation and integration. The basic learning outcomes include:
- Students understand the basic concepts in scientific computings,
including fixed-point numbers, floating-point numbers, single and double
precision, overflow and underflow, roundoff errors, absolute and relative
errors, stable and unstable computations.
- Students learn some basic knowledge in floating-point arithmetic and know
how to use the backward error analysis to estimate possible errors
generated in numerical computings.
- Students learn some basic concepts in solving nonlinear equations of one
variable: iterative methods, rate of convergence, local and global
convergence; learn some classical methods for solving nonlinear equations
of one variable: bisection methods, Newton's method, quasi-Newton's method
and fixed-point iteration, and learn how to analyse convergence of these
method.
- Students learn some basic methods for solving linear system of equations:
Cholesky factorization, LU and LDU factorization, forward- and
backward-substitutions for traingular systems, Gaussian elimination.
- Students learn sensitivity analysis of the solutions of linear systems
with respect to some errors in their coefficient matrices or in their
right-hand sides, and understand condition numbers and their importance.
- Students learn three polynomial interpolations, Newton form of
interpolations, Lagrange and spline interpolations, and learn how to
estimate the interpolation errors.
- Students learn some popular numerical integrations: trapezoidal and
composite trapezoidal rules, Simpson and composite Simpson rules,
Newton-Cotes quadrature rule, Gaussian quadrature rules, and learn how to
estimate the errors of quadrature rules.
- Students learn some basic numerical differentiation techniques: forward
differences, backward differences, central differences, and understand how
to estimate the errors of these difference schemes.
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