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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)only fo...
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 fo...
Newton-Gregory Forward Interpolation Formula
The Newton-Gregory Forward interpolation formula is a method used to approximate the value of a function within a given range. It is a form of polynomial interpolation and is based on the concept of divided differences.
Equally Spaced Intervals vs Unequally Spaced Intervals
Equally Spaced Intervals:
- Equally spaced intervals refer to a scenario where the difference between consecutive data points is constant.
- For example, if we have data points at x=0, x=1, x=2, x=3, and so on, the difference between any two consecutive data points is always 1.
- In this case, the interval between each data point is the same, and we can use the Newton-Gregory Forward interpolation formula.
Unequally Spaced Intervals:
- Unequally spaced intervals refer to a scenario where the difference between consecutive data points is not constant.
- For example, if we have data points at x=0, x=1, x=3, x=6, and so on, the difference between consecutive data points is not constant.
- In this case, the interval between each data point is not the same, and the Newton-Gregory Forward interpolation formula is not applicable.
Explanation:
The Newton-Gregory Forward interpolation formula is derived based on the assumption of equally spaced intervals. It utilizes divided differences to construct a polynomial that passes through the given data points.
The formula is given by:
f(x) ≈ f(x0) + (x - x0)f[x0, x1] + (x - x0)(x - x1)f[x0, x1, x2] + ...
where f[x0, x1] represents the divided difference of order 1, f[x0, x1, x2] represents the divided difference of order 2, and so on. These divided differences are calculated using the given data points.
Since the formula relies on the assumption of equally spaced intervals, it cannot be used for unequally spaced intervals. In the case of unequally spaced intervals, a different interpolation formula, such as the Lagrange interpolation formula or the Newton-Gregory Backward interpolation formula, should be used.
Therefore, the correct answer is option 'A' - the Newton-Gregory Forward interpolation formula can only be used for equally spaced intervals.
The Newton-Gregory Forward interpolation formula is a method used to approximate the value of a function within a given range. It is a form of polynomial interpolation and is based on the concept of divided differences.
Equally Spaced Intervals vs Unequally Spaced Intervals
Equally Spaced Intervals:
- Equally spaced intervals refer to a scenario where the difference between consecutive data points is constant.
- For example, if we have data points at x=0, x=1, x=2, x=3, and so on, the difference between any two consecutive data points is always 1.
- In this case, the interval between each data point is the same, and we can use the Newton-Gregory Forward interpolation formula.
Unequally Spaced Intervals:
- Unequally spaced intervals refer to a scenario where the difference between consecutive data points is not constant.
- For example, if we have data points at x=0, x=1, x=3, x=6, and so on, the difference between consecutive data points is not constant.
- In this case, the interval between each data point is not the same, and the Newton-Gregory Forward interpolation formula is not applicable.
Explanation:
The Newton-Gregory Forward interpolation formula is derived based on the assumption of equally spaced intervals. It utilizes divided differences to construct a polynomial that passes through the given data points.
The formula is given by:
f(x) ≈ f(x0) + (x - x0)f[x0, x1] + (x - x0)(x - x1)f[x0, x1, x2] + ...
where f[x0, x1] represents the divided difference of order 1, f[x0, x1, x2] represents the divided difference of order 2, and so on. These divided differences are calculated using the given data points.
Since the formula relies on the assumption of equally spaced intervals, it cannot be used for unequally spaced intervals. In the case of unequally spaced intervals, a different interpolation formula, such as the Lagrange interpolation formula or the Newton-Gregory Backward interpolation formula, should be used.
Therefore, the correct answer is option 'A' - the Newton-Gregory Forward interpolation formula can only be used for equally spaced intervals.
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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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