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0c0d3c84-c66e-416c-a8ab-86b7eb11dfb8 70 3 5 decimal

&[DATE] &[TIME] - &[FILENAME]

chr char ≡

sz size ≡

clr color ≡

vx vy chr sz clr plotG n vx length : plot vx 1 el vy 1 el chr sz clr augment : i 2 n range plot plot vx i el vy i el chr sz clr augment stack : for plot 41 line :

Low High step x fnct inc U Low High step range : i 1 U rows range f i el U i el fnct eval : for U f augment 31 line :

A robust construct of the Scale function M Low High Scale min M Min : max M Max : 12 mat maxmin max min - (1012 -^+ : HighLow High Low - (: i 1 M rows range j 1 M cols range N i j el M i j el min - maxmin / : for for HighLow N * Low + 51 line :

Fitting a polynomial to a data set Here, the polynomial is strictly Lagrangian, i.e: a sum of x^n monomials. It reproduces exactly the Mathcad built-in "regress". More interesting, the model φ(x) needs not be of integer monomials. This poly model is constructed manually [could not assign the symbolic]. Caution: if you encounter X's of very large values, the system will be numerically unstable. In this case, scale down the X's and rescale UP the final project. The regress-autoscale was introduced in Mathcad 11. Disambiguation: the PolyFit is a linear fit wrt the combination of Σx^n

t0 1 time :

Data 40 0.4 - 80 0.461 - 120 0.483 - 160 1.53 - 200 1.065 240 0.135 280 0.008 - 320 0.039 - 360 0.068 400 0.171 - 102 mat :

Prepare the Poly model

x φ 1 x x 2^x 3^41 mat :

1 x x 2^x 3^41 mat 0.070233 - 0.01244 - 0.00009 041 mat 21 mat

1 x x 1.75^x 2.5^41 mat 0.233002 0.025828 - 0.000706 0.000005 - 41 mat 21 mat

1 x x 1.75^x 2^41 mat 0.50361 0.040996 - 0.002156 0.000384 - 41 mat 21 mat

X Data 1 col : Y Data 2 col : 21 line

Expand a poly order

PolyOrder 5 :

x i^(i 0 PolyOrder sum 1 x 1 x 1 x + x 2^+ (* + x 4^+ (* +

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j 1 x φ length range i 1 X length range M i j el X i el φ j el : for for Φ M transpose M * : d M transpose Y * : b Φ 1 -^eval d * : 41 line

b 0.0702 - 0.0124 - 8.9683 105 -^* 1.4953 107 -^* - 41 mat

x fit b x φ * :

<= recast the scalar model

Smath QuickPlot scalar fit

x fit 20 x fit x diff ( * 2 1 sys

Note: Smath QuickPlot does not take mixed arguments,i.e: scalar fit(x) and data vector. Thus we construct the fit(x) in discrete form => Poly, so Smath is happy to plot both discrete arguments. The dscretiser inc(Low,High,step,fnct(x)) computes quick

Setup the plot

Poly 0 - 4501 x fit inc :

data X Y . 14 red plotG :

3rd order Lagrange PolyFit

Poly data 2 1 sys

Experiment x^n 1 x x 1.75 2 2.5 13 mat^x 3^41 mat

1 sec * 15 sec * # 31 mat

Get the roots from the fit

1 time t0 - 1.4 s *

r b polyroots (:

r 1 el fit r 2 el fit r 3 el fit 31 mat 6.285 1018 -^* - 2.1064 1015 -^* - 9.7862 1015 -^* 31 mat

b 0.070233331955239 - 0.012439727581565 - 8.96832805789608 105 -^* 1.49528739648005 107 -^* - 41 mat

r 5.43129355651816 - 231.271232761867 373.932925797578 31 mat

r fit vectorize 0.0702 - 0.0676 2.877 - 4.6516 - 31 mat 17.3395 0.157 6.6861 - 10.8105 - 31 mat 41 mat

1 x x 1.75^x 2.5^41 mat

<= Next: experiment some fractional monomials. Observe it takes lot more time to compute ... WHY ? Unlike integer monomials that take no time to compute, each fractional monomial consumes ~ 42 Aops [Aops = ArithmeticOperations]

Get more from this Smath tutorial document. Plug your new data set upstream the WS, construct/adapt the φ(x), run and get more kind of fit. Scale(M,Min,Max) will scale a matrix and/or a vector in the Min/Max range. You may need to scale the X data if they are too large, resulting in numerical instability.

Note that the Lagrange polynomial fit is same in all CAS [Computer Algebra System]. It is strictly based on the Cholesky matrix properties. Therefore much different than the NON-linear fit where a model function is wrt parameters β ... f(x,β) in which case the φ(x,β) model is the partial derivatives wrt to β. Cholesky solves directly in the Legendre least squares sense. With this tool, Lagrange interpolation is obsolete.

r 5.4313 - 231.2712 373.9329 31 mat :

The roots 'r' are the roots of fit(x) only. They can't be plugged in Lagrange, directly. Some scaling is required, maths not given].

x Lagrange x r 1 el - (x r 2 el - (* x r 3 el - (* :

Render Lagrange/roots ... brute force.
Observe numerical instability in the  'x' direction scale r 3 el r 2 el - 2 / r 2 el + fit r 3 el r 2 el - 2 / r 2 el + Lagrange / : 1.49528721921085 107 -^* -

scale x Lagrange * data 2 1 sys x fit scale x Lagrange * -

Note about polynomial approximations

The simple polynomial approximation to data set are generally limited to fit observed data, especially when the data are not regularly spaced. They may be adequte piecewise. High order polynomial become prone to instability and wiggling. If you figure you need a high order poly, then you need LinfitCheby. In Mathcad 11 we have used LinfitCheby up to order 99 no sweat. Another route is via Padé rational fractions. GenfitMatrix to Padé rational fraction does not implement easily in Smath . GenfitMatrix that generates Padé rational fraction is essentialy based on 'genfit', a built-in function in Mathcad 11. -------------------------------- All that said, there exist respectable but complicated poly in every day physics: Water Steam Properties IAPWS ... etc.

jmGiraud: 2015_07_15

PseudoCentroid of a real vector (V) V Psctrd i 1 V rows 1 - range W i el V i el V i el i i 1 + V rows sum * : for W W sum 12 mat 21 line :

r Psctrd 3287.04502833 - 86479.91050248 21 mat 83192.8654741501 12 mat

Export convenient

b 0.0702 - 0.0124 - 8.9683 105 -^* 1.4953 107 -^* - 41 mat :

x φ 1 x x 2^x 3^41 mat :

x fit b x φ * :

<= recast the scalar model

r b polyroots (11 line :

r 5.44492061378007 - 229.473926786148 375.736927085172 31 mat

x fit 5381469726562520000035162 x * + (* - 89683 x 2^* + (* 570411682128906 x 3^* - (* 19073486328125000000000 /

x q x fit 11 line :

x fnct 50 - x ≤ (x 450 ≤ (& x q out fit range if :

r 5.44492061378007 - 229.473926786148 375.736927085172 31 mat :

x fnct

r q vectorize 2.5273 1017 -^* - 2.7223 1016 -^* 2.45 1014 -^* 31 mat

r fit vectorize 0.0702 - 0.0675 2.8455 - 4.6591 - 31 mat 17.3865 0.1578 6.6522 - 10.8922 - 31 mat 41 mat