# Solve the Numerical Analysis Explain how Newtons interpolation formula better than the Lagrange formula.

Question
Polynomial arithmetic
Solve the Numerical Analysis Explain how Newtons interpolation formula better than the Lagrange formula.

2021-03-07
Lagrange interpolation is mostly just useful for theory. Actually computing with it requires huge numbers and catastrophic cancellations. In floating point arithmetic this is very bad. It does have some small advantages: for instance, the Lagrange approach amounts to diagonalizing the problem of finding the coefficients, so it takes only linear time to find the coefficients. This is good if you need to use the same set of points repeatedly. But all of these advantages do not make up for the problems associated with trying to actually evaluate a Lagrange interpolating polynomial.With Newton interpolation, you get the coefficients reasonably fast (quadratic time), the evaluation is much more stable (roughly because there is usually a single dominant term for a given x), the evaluation can be done quickly and straightforwardly using Horner's method, and adding an additional node just amounts to adding a single additional term. It is also fairly easy to see how to interpolate derivatives using the Newton framework.

### Relevant Questions

Solve the Numercial Method Explain how Newtons interpolation formula better tha the Lagrange formula
1. Explain with numerical examples what Real Numbers and Algebraic Expressions are. 2. Explain with numerical examples Factoring and finding LCMs (least common multiples). Explain factoring of a larger number. 3. Explain with numerical examples arithmetical operations (addition, subtraction, multiplication, division) with fractions 4, Explain with numerical examples arithmetical operations (addition, subtraction, multiplication, division) with percentages 5. Explain with numerical examples exponential notation 6. Explain with numerical examples order (precedence) of arithmetic operations 7. Explain with numerical examples the concept and how to find perimeter, area, volume, and circumference (use related formulas)

Consider the "clock arithmetic" group $$(Z_{15}, \oplus)$$ a) Using Lagrange`s Theotem, state all possible orders for subgroups of this group. b) List all of the subgroups of $$(Z_{15}, \oplus)$$

1. Is the sequence $$0.3, 1.2, 2.1, 3, ...$$ arithmetic? If so find the common difference.
2. An arithmetic sequence has the first term $$a_{1} = -4$$ and common difference $$d = - \frac{4}{3}$$. What is the $$6^{th}$$ term?
3. Write a recursive formula for the arithmetic sequence $$-2, - \frac{7}{2}, -5, - \frac{13}{2} ...$$ and then find the $$22^{nd}$$ term.
4. Write an explicit formula for the arithmetic sequence $$15.6, 15, 14.4, 13.8, ...$$ and then find the $$32^{nd}$$ term.
5. Is the sequence $$- 2, - 1, - \frac{1}{2},- \frac{1}{4},...$$ geometric? If so find the common ratio. If not, explain why.
A 1300-kg car coasts on a horizontal road, with a speed of18m/s. After crossing an unpaved sandy stretch of road 30.0 mlong, its speed decreases to 15m/s. If the sandy portion ofthe road had been only 15.0 m long, would the car's speed havedecreasedby 1.5 m/s, more than 1.5 m/s, or less than 1.5m/s?Explain. Calculate the change in speed in that case.
Consider the sequence $$67, 63, 59, 55......$$ a. show that the sequence is arithmetic. b. find a formula for the general term Un. c. Find the 60th term of the sequence. d. Is -143 a member of the sequence? e. Is 85 a member of the sequence?
For each of the following sequences: @ identify if the sequence is arithmetic, geometric or quadratic’. Justify your response. @ assuming the first item of each sequence is a1, give an expression for aj. (In other words, find a formula for the i-th term in the sequence). @ if the sequence is arithmetic or geometric, compute the sum of the first 10 terms in the sequence $$i 2,-12, 72, -432, 2592,...$$
$$ii 9, 18, 31, 48, 69, 94,...$$
$$iii 14, 11.5, 9, 6.5, 4, 1.5,...$$
Enzo's profit over the last few months has been failing. In December, he made $900. In January, his profit was$500, and in March, he lost \$300. is the sequence that represents his profit arithmetic? if it is, what is the recursive formula for the sequence?