Newtons Divided Difference Polynomial Method of Interpolation Mechanical
- Slides: 26
Newton’s Divided Difference Polynomial Method of Interpolation Mechanical Engineering Majors Authors: Autar Kaw, Jai Paul http: //numericalmethods. eng. usf. edu Transforming Numerical Methods Education for STEM Undergraduates http: //numericalmethods. eng. usf. edu 1
Newton’s Divided Difference Method of Interpolation http: //numericalmethods. eng. usf. edu
What is Interpolation ? Given (x 0, y 0), (x 1, y 1), …… (xn, yn), find the value of ‘y’ at a value of ‘x’ that is not given. 3 lmethods. eng. usf. edu http: //numerica
Interpolants Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate. 4 lmethods. eng. usf. edu http: //numerica
Newton’s Divided Difference Method Linear interpolation: Given linear interpolant through the data pass a where 5 lmethods. eng. usf. edu http: //numerica
Example A trunnion is cooled 80°F to − 108°F. Given below is the table of the coefficient of thermal expansion vs. temperature. Determine the value of the coefficient of thermal expansion at T=− 14°F using the direct method for linear interpolation. 6 Temperature (o. F) Thermal Expansion Coefficient (in/in/o. F) 80 6. 47 × 10− 6 0 6. 00 × 10− 6 − 60 5. 58 × 10− 6 − 160 4. 72 × 10− 6 − 260 3. 58 × 10− 6 − 340 2. 45 × 10− 6 lmethods. eng. usf. edu http: //numerica
Linear Interpolation 7 lmethods. eng. usf. edu http: //numerica
Linear Interpolation (contd) 8 lmethods. eng. usf. edu http: //numerica
Quadratic Interpolation 9 lmethods. eng. usf. edu http: //numerica
Example A trunnion is cooled 80°F to − 108°F. Given below is the table of the coefficient of thermal expansion vs. temperature. Determine the value of the coefficient of thermal expansion at T=− 14°F using the direct method for quadratic interpolation. 10 Temperature (o. F) Thermal Expansion Coefficient (in/in/o. F) 80 6. 47 × 10− 6 0 6. 00 × 10− 6 − 60 5. 58 × 10− 6 − 160 4. 72 × 10− 6 − 260 3. 58 × 10− 6 − 340 2. 45 × 10− 6 lmethods. eng. usf. edu http: //numerica
Quadratic Interpolation (contd) 11 lmethods. eng. usf. edu http: //numerica
Quadratic Interpolation (contd) 12 lmethods. eng. usf. edu http: //numerica
Quadratic Interpolation (contd) 13 lmethods. eng. usf. edu http: //numerica
General Form where Rewriting 14 lmethods. eng. usf. edu http: //numerica
General Form 15 lmethods. eng. usf. edu http: //numerica
General form 16 lmethods. eng. usf. edu http: //numerica
Example A trunnion is cooled 80°F to − 108°F. Given below is the table of the coefficient of thermal expansion vs. temperature. Determine the value of the coefficient of thermal expansion at T=− 14°F using the direct method for cubic interpolation. 17 Temperature (o. F) Thermal Expansion Coefficient (in/in/o. F) 80 6. 47 × 10− 6 0 6. 00 × 10− 6 − 60 5. 58 × 10− 6 − 160 4. 72 × 10− 6 − 260 3. 58 × 10− 6 − 340 2. 45 × 10− 6 lmethods. eng. usf. edu http: //numerica
Example 18 lmethods. eng. usf. edu http: //numerica
Example 19 lmethods. eng. usf. edu http: //numerica
Example 20 lmethods. eng. usf. edu http: //numerica
Comparison Table 21 lmethods. eng. usf. edu http: //numerica
Reduction in Diameter The actual reduction in diameter is given by where Tr = room temperature (°F) Tf = temperature of cooling medium (°F) Since Tr = 80 °F and Tr = − 108 °F, Find out the percentage difference in the reduction in the diameter by the above integral formula and the result using thermal expansion coefficient from the cubic interpolation. 22 lmethods. eng. usf. edu http: //numerica
Reduction in Diameter We know from interpolation that Therefore, 23 lmethods. eng. usf. edu http: //numerica
Reduction in diameter Using the average value for the coefficient of thermal expansion from cubic interpolation The percentage difference would be 24 lmethods. eng. usf. edu http: //numerica
Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, Math. Cad and MAPLE, blogs, related physical problems, please visit http: //numericalmethods. eng. usf. edu/topics/newton_div ided_difference_method. html
THE END http: //numericalmethods. eng. usf. edu
- Spline interpolation vs polynomial interpolation
- Lagrange interpolation formula
- (x+1)^3 formula
- Neville's algorithm example
- Polynomial interpolation
- Matplotlib inline
- How to divide a polynomial by another polynomial
- Newtons method matlab
- Finite differences and interpolation
- Stirling formula in numerical analysis
- Interpolation between two points
- Direct method of interpolation
- Bicubic interpolation
- Direct method interpolation
- Direct interpolation method
- Quadratic extrapolation
- Lagrange method interpolation
- Inverse parabolic interpolation
- Actual mechanical advantage vs ideal mechanical advantage
- Finite divided difference
- Finite difference equation
- Finite divided difference
- Difference between powders and granules
- Problems encountered in powder formulation
- Finely divided, bulk effervescent
- Dry gum and wet gum method is used to prepare
- Difference between repetition method and reiteration method