Numerical analysis
- 1. NUMERICAL ANALYSIS (15UMTC64) III B.SC. MATHEMATICS (SF) Ms. M. Mohanamalar, M.Sc., M.Phil., Ms. N. Malathi, M.Sc.,
- 2. • Calculus is the mathematics of change. Because engineers must continuously deal with systems and processes that change, calculus is an essential tool of engineering. • Standing in the heart of calculus are the mathematical concepts of differentiation and integration • Numerical differentiation is the process of finding the numerical value of the derivative of a given function at a given point.
- 3. For some particular value of x from the given data (xi,yi), i=1, 2, 3…..n where y=f(x) i explicitly, the interpolation to be used depends on the particular value of x which derivatives are required. •If the values of x are not equally spaced, we represent the function by difference formula and the derivatives are obtained. •If the values of x are equally spaced, the derivatives are calculated by using Newton’s Forward interpolation formula or backward interpolation formula.
- 4. Principle: First fit a polynomial for the given difference data interpolation using Newton ’s divided difference interpolation formula and compute the derivatives for a given x.
- 5. Example
- 6. Example
- 7. erivatives using Finite difference Newton’s Forward Difference formula is Newton’s Backward Difference formula is
- 8. erivatives using Finite difference
- 9. erivatives using Finite difference
- 10. erivatives using Finite difference Example 2. Find the first two derivatives of y at x=54 from the following data x 50 51 52 53 54 y 3.6840 3.7083 3.7325 3.7563 3.7798
- 11. erivatives using Finite difference Example
- 12. erivatives using Finite difference Example 1 Find first and second derivatives of the function at the point x=1.2 from the following data x 1 2 3 4 5 y 0 1 5 6 8 The difference table
- 13. erivatives using Finite difference Example