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Newton- Gregory Forward interpolation formula can be used _______
- a)only for equally spaced intervals
- b)only for unequally spaced intervals
- c)for both equally and unequally spaced intervals
- d)for unequally intervals
Correct answer is option 'A'. Can you explain this answer?
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Newton- Gregory Forward interpolation formula can be used _______a)onl...
Newton-Gregory Forward Interpolation Formula
The Newton-Gregory Forward Interpolation Formula is a method used to approximate the value of a function at a given point within a specified range. It is a polynomial interpolation method that can be used when the interval between the data points is equal.
Equally Spaced Intervals
The Newton-Gregory Forward Interpolation Formula is specifically designed for equally spaced intervals. In this context, equally spaced intervals refer to a set of data points that are uniformly distributed along the x-axis. The formula allows us to estimate the value of a function at any point within the given interval.
Formula and Procedure
The Newton-Gregory Forward Interpolation Formula can be expressed as:
f(x) = f(x0) + (x - x0)Δf(x0) + (x - x0)(x - x1)Δ²f(x0) + ... + (x - x0)(x - x1)...(x - xn-1)Δⁿf(x0)
where:
- f(x) represents the value of the function at the desired point x.
- f(x0), f(x1), ..., f(xn) are the given data points.
- Δf(x0), Δ²f(x0), ..., Δⁿf(x0) are the forward differences at x0.
The procedure to apply the Newton-Gregory Forward Interpolation Formula involves the following steps:
1. Organize the given data points in ascending order based on the x-values.
2. Calculate the forward differences using the formula:
Δf(x0) = f(x1) - f(x0)
Δ²f(x0) = Δf(x1) - Δf(x0)
Δ³f(x0) = Δ²f(x1) - Δ²f(x0)
...
Δⁿf(x0) = Δⁿ⁻¹f(x1) - Δⁿ⁻¹f(x0)
3. Substitute the values into the formula and simplify to find the estimated value of the function at the desired point x.
Conclusion
In conclusion, the Newton-Gregory Forward Interpolation Formula is applicable only for equally spaced intervals. It provides an efficient method to approximate the value of a function at any point within the given interval. The formula relies on the concept of forward differences, which are calculated based on the differences between consecutive data points. Therefore, it is important to ensure that the data points are equally spaced for accurate results.
The Newton-Gregory Forward Interpolation Formula is a method used to approximate the value of a function at a given point within a specified range. It is a polynomial interpolation method that can be used when the interval between the data points is equal.
Equally Spaced Intervals
The Newton-Gregory Forward Interpolation Formula is specifically designed for equally spaced intervals. In this context, equally spaced intervals refer to a set of data points that are uniformly distributed along the x-axis. The formula allows us to estimate the value of a function at any point within the given interval.
Formula and Procedure
The Newton-Gregory Forward Interpolation Formula can be expressed as:
f(x) = f(x0) + (x - x0)Δf(x0) + (x - x0)(x - x1)Δ²f(x0) + ... + (x - x0)(x - x1)...(x - xn-1)Δⁿf(x0)
where:
- f(x) represents the value of the function at the desired point x.
- f(x0), f(x1), ..., f(xn) are the given data points.
- Δf(x0), Δ²f(x0), ..., Δⁿf(x0) are the forward differences at x0.
The procedure to apply the Newton-Gregory Forward Interpolation Formula involves the following steps:
1. Organize the given data points in ascending order based on the x-values.
2. Calculate the forward differences using the formula:
Δf(x0) = f(x1) - f(x0)
Δ²f(x0) = Δf(x1) - Δf(x0)
Δ³f(x0) = Δ²f(x1) - Δ²f(x0)
...
Δⁿf(x0) = Δⁿ⁻¹f(x1) - Δⁿ⁻¹f(x0)
3. Substitute the values into the formula and simplify to find the estimated value of the function at the desired point x.
Conclusion
In conclusion, the Newton-Gregory Forward Interpolation Formula is applicable only for equally spaced intervals. It provides an efficient method to approximate the value of a function at any point within the given interval. The formula relies on the concept of forward differences, which are calculated based on the differences between consecutive data points. Therefore, it is important to ensure that the data points are equally spaced for accurate results.
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Newton- Gregory Forward interpolation formula can be used _______a)onl...
Newton – Gregory Forward Interpolation formula is given by
f(x) = y0 + nΔy0 + n(n-1)Δ2y0/2! + n(n-1)(n-2) Δ3y0 /3! + …..
This formula is obtained by the Newton’s Divided difference formula by substituting the intervals as h. This is done because we assume the intervals to be constant, that is, equally spaced.
f(x) = y0 + nΔy0 + n(n-1)Δ2y0/2! + n(n-1)(n-2) Δ3y0 /3! + …..
This formula is obtained by the Newton’s Divided difference formula by substituting the intervals as h. This is done because we assume the intervals to be constant, that is, equally spaced.
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Newton- Gregory Forward interpolation formula can be used _______a)only for equally spaced intervalsb)only for unequally spaced intervalsc)for both equally and unequally spaced intervalsd)for unequally intervalsCorrect answer is option 'A'. Can you explain this answer?
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