[HN Gopher] Notes on Lagrange Interpolating Polynomials
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       Notes on Lagrange Interpolating Polynomials
        
       Author : ibobev
       Score  : 34 points
       Date   : 2026-03-02 16:01 UTC (7 hours ago)
        
 (HTM) web link (eli.thegreenplace.net)
 (TXT) w3m dump (eli.thegreenplace.net)
        
       | wolfi1 wrote:
       | the last matrix before the appendix is not the identity matrix,
       | right now the matrix is: \begin{bmatrix} 1 & 0 & 0 &
       | \dots & 0\\\ 1 & 0 & 0 & \dots & 0\\\ 1 &
       | 0 & 0 & \dots & 0\\\ \vdots & \vdots & \vdots
       | & \ddots &\vdots \\\ 1 & 0 & 0 & \dots &
       | 1 \end{bmatrix}
        
         | eliben wrote:
         | Thanks for noticing, I'll fix it shortly
        
       | hdrz wrote:
       | The Lagrange polynomials form the normal basis of most Finite
       | Elements Method (FEM) software. There are other polynomials which
       | are used as well, but these are the workhorse of most solvers.
        
       | commandersaki wrote:
       | In the Polynomial Interpolation Theorem, you have r(x) = p(x) -
       | r(x), but I think it should be q(x) = p(x) - r(x).
        
         | eliben wrote:
         | Fixed, thank you! (it's actually r(x)=p(x)-q(x))
         | 
         | (proof-reading through HN is a mildly embarrassing process,
         | sorry about that! I do go over these posts and proof-read them
         | several times myself before publishing)
        
       | wenc wrote:
       | I used to solve differential algebraic equations using Lagrange
       | polynomials.
       | 
       | Essentially you convert the differential equations into an
       | algebraic system by discretizing the solution. The method is
       | called Orthogonal Collocation on Finite Elements (OCFE), and it
       | was developed by chemical engineers.
       | 
       | The Lagrange polynomials were calculated at special knots that
       | corresponded to Radau interior points, which work great for stiff
       | systems.
       | 
       | It's great for solving differential algebraic equations through
       | purely sparse matrix operations, no explicit integration like
       | Runge Kutta. (Well, it's implicit Runge Kutta).
        
       | TimorousBestie wrote:
       | If you're interested in numerical stability issues associated
       | with Lagrange interpolation, Trefethen and Boyd is still
       | relevant:
       | 
       | https://people.maths.ox.ac.uk/trefethen/barycentric.pdf
        
       | vector_spaces wrote:
       | Every polynomial interpolates itself -- meaning that you can
       | often apply this interpolation procedure to your favorite/nemesis
       | polynomial or equivalently rewrite your polynomial of interest in
       | this Lagrange basis, and see if the coefficients lead you
       | anywhere. This is especially helpful in proving polynomial
       | inequalities. For instance, Chebyshev polynomials T_n enjoy an
       | alternation property over their extremal points -- so in the
       | Lagrange basis, in many problems (e.g. Markov type inequalities)
       | they emerge as the extremal case in the triangle inequality.
       | 
       | My beef with this approach is that it is a little unsatisfying in
       | the sense that it sort of feels like we "got lucky". That is, it
       | highlights this special feature (alternation) while burying the
       | interesting structure that leads to such polynomials being
       | extremal in these problems, as can be seen if you attempt certain
       | seemingly trivial extensions of classical inequalities -- but
       | nevertheless it's an important trick in extremal polynomial
       | theory and approximation more broadly
        
       | 011101101 wrote:
       | \begin{pmatrix} H_{1} & H_{2} &\dots & P_{Mc} V_{W} & V_{1}
       | &\dots & P_{Mc} \hdotsfor{4} \\\ \end{pmatrix}
        
       | looneysquash wrote:
       | Maybe there's a gap in my knowledge of notation, but I was
       | confused by:
       | 
       | > Let's define the Lagrange basis functions
       | 
       | It looks like you defined `l_i(x)` as a piecewise function or a
       | step function.
       | 
       | But then you show with it's actual definition later in that
       | section. (That's what that section is building to and
       | explaining.)
        
       | drivebyhooting wrote:
       | If we already have access the machinery of the fundamental
       | theorem of algebra then the invertibility of the Vandermonde
       | Matrix follows as a corollary.
        
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