Thomas write rule

In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points $x_j$ and numbers $y_j$, the Lagrange polynomial is the polynomial of the least degree that at each point $x_j$ assumes the corresponding value $y_j$ (i.e. the functions coincide at each point). The interpolating polynomial of the least degree is unique, however, and it is therefore more appropriate to speak of "the Lagrange form" of that unique polynomial rather than "the Lagrange interpolation polynomial", since the same polynomial can be arrived at through multiple methods. Although named after Joseph Louis Lagrange, who published it in 1795, it was first discovered in 1779 by Edward Waring and it is also an easy consequence of a formula published in 1783 by Leonhard Euler.

Lagrange interpolation is susceptible to Runge's phenomenon, and the fact that changing the interpolation points requires recalculating the entire interpolant can make Newton polynomials easier to use. Lagrange polynomials are used in the Newton–Cotes method of numerical integration and in Shamir's secret sharing scheme in cryptography.

This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L(x) (in black), which is the sum of the scaled basis polynomials y₀ℓ₀(x), y₁ℓ₁(x), y₂ℓ₂(x) and y₃ℓ₃(x). The interpolation polynomial passes through all four control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points.

Definition

Given a set of k + 1 data points

$(x_0,y_0),\dots ,(x_j,y_j),\dots ,(x_k,y_k)$

where no two $x_j$ are the same, the interpolation polynomial in the Lagrange form is a linear combination

$L(x):=\sum _{j=0}^k{y}_j{\ell }_j(x)$

of Lagrange basis polynomials

${\ell }_j(x):=\prod _{\begin{array}{c}0\le m\le k\\ m\ne j\end{array}}\frac{x-x_m}{x_j-x_m}=\frac{(x-x_0)}{(x_j-x_0)}\cdots \frac{(x-x_{j-1})}{(x_j-x_{j-1})}\frac{(x-x_{j+1})}{(x_j-x_{j+1})}\cdots \frac{(x-x_k)}{(x_j-x_k)},$

where $0\le j\le k$. Note how, given the initial assumption that no two $x_i$ are the same, $x_j-x_m\ne 0$, so this expression is always well-defined. The reason pairs $x_i=x_j$ with $y_i\ne y_j$ are not allowed is that no interpolation function $L$ such that $y_i=L(x_i)$ would exist; a function can only get one value for each argument $x_i$. On the other hand, if also $y_i=y_j$, then those two points would actually be one single point.

For all $j\ne i$, ${\ell }_j(x)$ includes the term $(x-x_i)$ in the numerator, so the whole product will be zero at $x=x_i$:

${\ell }_{j\ne i}(x_i)=\prod _{m\ne j}\frac{x_i-x_m}{x_j-x_m}=\frac{(x_i-x_0)}{(x_j-x_0)}\cdots \frac{(x_i-x_i)}{(x_j-x_i)}\cdots \frac{(x_i-x_k)}{(x_j-x_k)}=0.$

On the other hand,

${\ell }_i(x_i):=\prod _{m\ne i}\frac{x_i-x_m}{x_i-x_m}=1$

In other words, all basis polynomials are zero at $x=x_i$, except ${\ell }_i(x)$, for which it holds that ${\ell }_i(x_i)=1$, because it lacks the $(x-x_i)$ term.

It follows that $y_i{\ell }_i(x_i)=y_i$, so at each point $x_i$, $L(x_i)=y_i+0+0+\dots +0=y_i$, showing that $L$ interpolates the function exactly.

Proof

The function L(x) being sought is a polynomial in $x$ of the least degree that interpolates the given data set; that is, assumes value $y_j$ at the corresponding $x_j$ for all data points $j$:

$L(x_j)=y_{j}j=0,\dots ,k$

Observe that:

1. In ${\ell }_j(x)$ there are k terms in the product and each term contains one x, so L(x) (which is a sum of these k-degree polynomials) must also be a k-degree polynomial.
2. ${\ell }_j(x_i)=\prod _{m=0,m\ne j}^k\frac{x_i-x_m}{x_j-x_m}$

We consider what happens when this product is expanded. Because the product skips $m=j$, if $i=j$ then all terms are $\frac{x_j-x_m}{x_j-x_m}=1$ (except where $x_j=x_m$, but that case is impossible, as pointed out in the definition section—in that term, $m=j$, and since $m\ne j$, $i\ne j$, contrary to $i=j$). Also if $i\ne j$ then since $m\ne j$ does not preclude it, one term in the product will be for $m=i$, i.e. $\frac{x_i-x_i}{x_j-x_i}=0$, zeroing the entire product. So

1. ${\ell }_j(x_i)={\delta }_{ji}=\{\begin{array}{cc}1,& \text{if }j=i\\ 0,& \text{if }j\ne i\end{array}$

where ${\delta }_{ij}$ is the Kronecker delta. So:

$L(x_i)=\sum _{j=0}^k{y}_j{\ell }_j(x_i)=\sum _{j=0}^k{y}_j{\delta }_{ji}=y_i.$

Thus the function L(x) is a polynomial with degree at most k and where $L(x_i)=y_i$.

Additionally, the interpolating polynomial is unique, as shown by the unisolvence theorem at polynomial interpolation article.

Main idea

Solving an interpolation problem leads to a problem in linear algebra amounting to inversion of a matrix. Using a standard monomial basis for our interpolation polynomial Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

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A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. for the coefficients Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

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Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., we merely get the identity matrix, δ_ij, which is its own inverse: the Lagrange basis automatically inverts the analog of the Vandermonde matrix.

This construction is analogous to the Chinese Remainder Theorem. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears.

Examples

Example 1

The tangent function and its interpolant

Find an interpolation formula for ƒ(x) = tan(x) given this set of known values:

$\begin{array}{cccccccc}{x}_0& =-1.5& & & & & f(x_0)& =-14.1014\\ x_1& =-0.75& & & & & f(x_1)& =-0.931596\\ x_2& =0& & & & & f(x_2)& =0\\ x_3& =0.75& & & & & f(x_3)& =0.931596\\ x_4& =1.5& & & & & f(x_4)& =\mathrm{14.1014.}\end{array}$

The Lagrange basis polynomials are:

${\ell }_0(x)=\frac{x-x_1}{x_0-x_1}\cdot \frac{x-x_2}{x_0-x_2}\cdot \frac{x-x_3}{x_0-x_3}\cdot \frac{x-x_4}{x_0-x_4}=\frac{1}{243}x(2x-3)(4x-3)(4x+3)$

${\ell }_1(x)=\frac{x-x_0}{x_1-x_0}\cdot \frac{x-x_2}{x_1-x_2}\cdot \frac{x-x_3}{x_1-x_3}\cdot \frac{x-x_4}{x_1-x_4}=-\frac{8}{243}x(2x-3)(2x+3)(4x-3)$

${\ell }_2(x)=\frac{x-x_0}{x_2-x_0}\cdot \frac{x-x_1}{x_2-x_1}\cdot \frac{x-x_3}{x_2-x_3}\cdot \frac{x-x_4}{x_2-x_4}=\frac{3}{243}(2x+3)(4x+3)(4x-3)(2x-3)$

${\ell }_3(x)=\frac{x-x_0}{x_3-x_0}\cdot \frac{x-x_1}{x_3-x_1}\cdot \frac{x-x_2}{x_3-x_2}\cdot \frac{x-x_4}{x_3-x_4}=-\frac{8}{243}x(2x-3)(2x+3)(4x+3)$

${\ell }_4(x)=\frac{x-x_0}{x_4-x_0}\cdot \frac{x-x_1}{x_4-x_1}\cdot \frac{x-x_2}{x_4-x_2}\cdot \frac{x-x_3}{x_4-x_3}=\frac{1}{243}x(2x+3)(4x-3)(4x+3).$

Thus the interpolating polynomial then is

$\begin{array}{cc}L(x)& =\frac{1}{243}(f(x_0)x(2x-3)(4x-3)(4x+3)\\ & -8f(x_1)x(2x-3)(2x+3)(4x-3)\\ & +3f(x_2)(2x+3)(4x+3)(4x-3)(2x-3)\\ & -8f(x_3)x(2x-3)(2x+3)(4x+3)\\ & +f(x_4)x(2x+3)(4x-3)(4x+3))\\ & =4.834848x^3-1.477474x.\end{array}$

Example 2

We wish to interpolate ƒ(x) = x² over the range 1 ≤ x ≤ 3, given these three points:

$\begin{array}{cccccc}{x}_0& =1& & & f(x_0)& =1\\ x_1& =2& & & f(x_1)& =4\\ x_2& =3& & & f(x_2)& =9.\end{array}$

The interpolating polynomial is:

$\begin{array}{cc}L(x)& =1\cdot \frac{x-2}{1-2}\cdot \frac{x-3}{1-3}+4\cdot \frac{x-1}{2-1}\cdot \frac{x-3}{2-3}+9\cdot \frac{x-1}{3-1}\cdot \frac{x-2}{3-2}\\ & =x^2.\end{array}$

Example 3

We wish to interpolate ƒ(x) = x³ over the range 1 ≤ x ≤ 3, given these 3 points:

$x_0=1$	$f(x_0)=1$
$x_1=2$	$f(x_1)=8$
$x_2=3$	$f(x_2)=27$

The interpolating polynomial is:

$\begin{array}{cc}L(x)& =1\cdot \frac{x-2}{1-2}\cdot \frac{x-3}{1-3}+8\cdot \frac{x-1}{2-1}\cdot \frac{x-3}{2-3}+27\cdot \frac{x-1}{3-1}\cdot \frac{x-2}{3-2}\\ & =6x^2-11x+6.\end{array}$

Notes

Example of Lagrange polynomial interpolation divergence.

The Lagrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial. Therefore, it is preferred in proofs and theoretical arguments. Uniqueness can also be seen from the invertibility of the Vandermonde matrix, due to the non-vanishing of the Vandermonde determinant.

But, as can be seen from the construction, each time a node x_k changes, all Lagrange basis polynomials have to be recalculated. A better form of the interpolation polynomial for practical (or computational) purposes is the barycentric form of the Lagrange interpolation (see below) or Newton polynomials.

Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon; the problem may be eliminated by choosing interpolation points at Chebyshev nodes.

The Lagrange basis polynomials can be used in numerical integration to derive the Newton–Cotes formulas.

Barycentric interpolation

Using

$\ell (x)=(x-x_0)(x-x_1)\cdots (x-x_k)$

we can rewrite the Lagrange basis polynomials as

${\ell }_j(x)=\frac{\ell (x)}{x-x_j}\frac{1}{\prod _{i=0,i\ne j}^k(x_j-x_i)}$

or, by defining the barycentric weights

$w_j=\frac{1}{\prod _{i=0,i\ne j}^k(x_j-x_i)}$

we can simply write

${\ell }_j(x)=\ell (x)\frac{w_j}{x-x_j}$

which is commonly referred to as the first form of the barycentric interpolation formula.

The advantage of this representation is that the interpolation polynomial may now be evaluated as

$L(x)=\ell (x)\sum _{j=0}^k\frac{w_j}{x-x_j}{y}_j$

which, if the weights $w_j$ have been pre-computed, requires only $𝒪(n)$ operations (evaluating $\ell (x)$ and the weights $w_j/(x-x_j)$) as opposed to $𝒪(n^2)$ for evaluating the Lagrange basis polynomials ${\ell }_j(x)$ individually.

The barycentric interpolation formula can also easily be updated to incorporate a new node $x_{k+1}$ by dividing each of the $w_j$, $j=0\dots k$ by $(x_j-x_{k+1})$ and constructing the new $w_{k+1}$ as above.

We can further simplify the first form by first considering the barycentric interpolation of the constant function $g(x)\equiv 1$:

$g(x)=\ell (x)\sum _{j=0}^k\frac{w_j}{x-x_j}.$

Dividing $L(x)$ by $g(x)$ does not modify the interpolation, yet yields

$L(x)=\frac{\sum _{j=0}^k\frac{w_j}{x-x_j}{y}_j}{\sum _{j=0}^k\frac{w_j}{x-x_j}}$

which is referred to as the second form or true form of the barycentric interpolation formula. This second form has the advantage that $\ell (x)$ need not be evaluated for each evaluation of $L(x)$.

Finite fields

The Lagrange polynomial can also be computed in finite fields. This has applications in cryptography, such as in Shamir's Secret Sharing scheme.

See also

• Polynomial interpolation
• Neville's algorithm
• Newton form of the interpolation polynomial
• Bernstein form of the interpolation polynomial
• Newton–Cotes formulas
• Carlson's theorem
• Lebesgue constant (interpolation)
• The Chebfun system
• Table of Newtonian series
• Frobenius covariant
• Sylvester's formula

References

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• ALGLIB has an implementations in C++ / C# / VBA / Pascal.
• GSL has a polynomial interpolation code in C
• Lagrange Method of Interpolation — Notes, PPT, Mathcad, Mathematica, MATLAB, Maple at Holistic Numerical Methods Institute
• Lagrange interpolation polynomial on www.math-linux.com
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• Module for Lagrange Polynomials by John H. Mathews
• Dynamic Lagrange interpolation with JSXGraph
• Numerical computing with functions: The Chebfun Project
• Excel Worksheet Function for Bicubic Lagrange Interpolation