Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.

Sampling a polynomial of degree k − 1 at more than k points creates an overdetermined system, and allows recovery of the polynomial at the receiver given any k out of n sample points using ( Lagrange ) interpolation.

( proves this last fact using Lagrange interpolation.

Therefore the Newton form of the interpolation polynomial is usually preferred over the Lagrange form for practical purposes, although, in actual fact ( and contrary to widespread claims ), Lagrange, too, allows calculation of the next higher degree interpolation without re-doing previous calculations — and is considerably easier to evaluate.

In numerical analysis, Lagrange polynomials are used for polynomial interpolation.

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.

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

One can use Lagrange polynomial interpolation to find an expression for this polynomial,

Let L ( x ) be the interpolation polynomial in the Lagrange form for the given data points, then

A more direct computation, which is strongly related with Lagrange interpolation consists in writing

Let and let be n + 1 distinct elements of K. Then for and by Lagrange interpolation we have.

In particular, the roots of are simple, and the " interpolation " characterization tells us that is given by the Lagrange interpolation formula.

Applying the Lagrange interpolation formula to do quadratic interpolation on the inverse of f yields

Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines, or cubic convolution algorithm.

In contrast to Lagrange interpolation and Hermite interpolation, a Birkhoff interpolation problem does not always have a unique solution.

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

Lagrange proved that for any natural number n that is not a perfect square there are x and y > 0 that satisfy Pell's equation.

This algebra is quotiented over by the ideal generated by the Euler – Lagrange equations.

Writing coordinates ( t, x ) = ( x < sup > 0 </ sup >, x < sup > 1 </ sup >, x < sup > 2 </ sup >, x < sup > 3 </ sup >) = x < sup > μ </ sup >, this form of the Euler – Lagrange equation is

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order ( number of elements ) of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange.

By using Lagrange multipliers and seeking the extremum of the Lagrangian, it may be readily shown that the solution to the equality constrained problem is given by the linear system:

where is a set of Lagrange multipliers which come out of the solution alongside.

* It was proven by Lagrange that every positive integer is the sum of four squares.

The technology is practical and is either derived from true science ( such as Lagrange points in space and the O ' Neill cylinder as a living environment ) or at least well-explained, feasible technology, requiring only a few fictional elements to function ( such as Minovsky Physics as a means of energy production from helium-3 ).

Not all of the ring material would have necessarily been swept up right away ; the thickened crust of the Far Side suggests that a second moon about 1, 000-km in diameter formed in a Lagrange point of the Moon ; after tens of millions of years, as the two moons migrated outward from the Earth, solar tidal effects would have made the Lagrange orbit unstable, resulting in a slow-velocity collision that would have ' pancaked ' the smaller moon onto what is now the Far Side.

The Sun-Jupiter Trojan asteroid system is an example of a stable Lagrange orbit.

If a star grows outside of its Roche lobe too fast for all abundant matter to be transferred to the other component, it is also possible that matter will leave the system through other Lagrange points or as stellar wind, thus being effectively lost to both components.

In classical mechanics, the equation of motion is Newton's second law, and equivalent formulations are the Euler – Lagrange equations and Hamilton's equations.

In the case when the coefficient ring is a field of characteristic 0, the Lagrange inversion formula provides a powerful tool to compute the coefficients of g, as well as the coefficients of the ( multiplicative ) powers of g.

where f is a known power series with f ( 0 ) = 0, then the coefficients of g can be explicitly computed using the Lagrange inversion formula.

Since the proof of the Lagrange inversion formula is a very short computation, it is worth reporting it here.

One may observe that the above computation can be repeated plainly in more general settings than: a generalization of the Lagrange inversion formula is already available working in the-modules, where is a complex exponent.

The key advantage of a linear penalty function is that the slack variables vanish from the dual problem, with the constant C appearing only as an additional constraint on the Lagrange multipliers.

The Taylor series of around 0 can be found using the Lagrange inversion theorem and is given by

In this context, multipliers are unrelated to Lagrange multipliers, except for the fact that they both involve the multiplication operation.

This follows from the fact that at the optimal solution the gradient of the objective function is a linear combination of the constraint function gradients with the weights equal to the Lagrange multipliers.

It is suspected that many Trojan asteroids are in fact small planetesimals captured in the Lagrange point of Jupiter – Sun system during the outer migration of the giant planets, 3. 9 billion years ago.

In the distant future, Mankind has colonized space ( with clusters of space colonies at each of the five Earth-Moon Lagrange points ), and, down on the Earth, the nations have united as the United Earth Sphere Alliance.

Paths have been calculated which link the Lagrange points of the various planets into the so-called Interplanetary Transport Network.

The arrows indicate the gradients of the potential around the five Lagrange points — downhill toward them (< span style =" color: red ;"> red </ span >) or away from them (< span style =" color: blue ;"> blue </ span >).

The Lagrangian points (; also Lagrange points, L-points, or libration points ) are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be part of a constant-shape pattern with two larger objects ( such as a satellite with respect to the Earth and Moon ).

The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to orbit with them.

In 1772, Italian-born mathematician Joseph-Louis Lagrange, in studying the restricted three-body problem, predicted that a small body sharing an orbit with a planet but lying 60 ° ahead or behind it will be trapped near these points.

These leading and trailing points are called the and Lagrange points.

However, no asteroids trapped in Lagrange points were observed until more than a century after Lagrange's hypothesis.

Most inner moons of planets have synchronous rotation, so their synchronous orbits are, in practice, limited to their leading and trailing ( and ) Lagrange points, as well as the and Lagrange points, assuming they do not fall within the body of the moon.

Supporting this theory, extrasolar planets have been discovered in Lagrange points of each other, and are expected to collide, after co-orbiting for millions of years.

After Newton, Lagrange ( 25 January 1736 – 10 April 1813 ) attempted to solve the three-body problem, analyzed the stability of planetary orbits, and discovered the existence of the Lagrangian points.

If is a maximum of for the original constrained problem, then there exists such that is a stationary point for the Lagrange function ( stationary points are those points where the partial derivatives of are zero, i. e. ).

Lagrange multipliers cause the critical points to occur at saddle points.