82930fbf-7acb-44c4-b396-d6a9d167a108 10 4 5 decimal

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The 3 types of "cubic spline"

Essentially, the coefficients of the cubic spline consists in solving the basis tridiagonal matrix wrt the 'br' vector of derivatives . Independently of the type of cubic spline [type: 0, 1, 2 = l, p, c] two steps of preparation are needed: 1. Install the constant vector 'br' 2. Install the blank tridiagonal matrix.

XY 3.52 - 2.358 - 3 - 2.213 - 2 - 1.256 - 1 - 1.739 - 0 1.482 - 1 1.277 - 62 mat :

X XY 1 col :

Y XY 2 col :

N Y rows 1 - :

N 5

r 2 N range :

Prep.0 Install the constant vector 'br'

i r br i el Y i 1 + el Y i el - X i 1 + el X i el - / (Y i el Y i 1 - el - X i el X i 1 - el - / (- : br N 1 + el 0 : 21 line for

br 0 0.6782 1.44 - 0.74 0.052 - 061 mat

Prep.1 Install the tridiagonal matrix

Initialise the blank
diagonal matrix M 11 el 1 : M N 1 + N 1 + el 1 : 21 sys

M 10000000000000000000000000000000000166 mat

Observe the population of the rows upper left row ---------------------- ----------------------lower right row respectively to linear/parabolic/cubic. The basis tridiagonal is the "linear", further modified to suit user l_p_c .

0. Construct the tridiagonal linear spline matrix [lspline]

i r M i i 1 - el X i el X i 1 - el - 6 / : for

i r M i i el X i 1 + el X i 1 - el - 3 / : for

i r M i i 1 + el X i 1 + el X i el - 6 / : for

s M 1 -^br * :

M 100000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00000166 mat

s 0 2.419 3.285 - 2.0808 0.5982 - 061 mat

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

s M 1 -^br * :

M 11 - 0000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00001 - 166 mat

s 2.0344 2.0344 3.1736 - 2.02 0.4664 - 0.4664 - 61 mat

2. Construct the tridiagonal cubic spline matrix [cspline]

M 12 el X 1 el X 3 el - (X 3 el X 2 el - / (:

M N 1 + N el X N el X N 2 - el - (X N 2 - el X N 1 - el - / :

M 13 el X 2 el X 1 el - (X 3 el X 2 el - (/ :

M N 1 + N 1 - el X N 1 - el X N el - (X N 1 - el X N el - (/ :

s M 1 -^br * :

M 1 1.52 - 0.52 000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00012 - 166 mat

s 7.1661 1.0623 2.8867 - 1.8447 0.052 - 1.9487 - 61 mat

Construct the general spline cubic polynomial

s 0 2.419 3.285 - 2.0808 0.5982 - 061 mat :

In there, plug the 's' of the selected type of spline. In this demo, we select "linear" type in order to view all segments on the XY data set

Linear 0 2.419 3.285 - 2.0808 0.5982 - 061 mat

Parabolic 2.0344 2.0344 3.1736 - 2.02 0.4664 - 0.4664 - 61 mat

Cubic 4.0766 1.6415 3.0412 - 1.8833 0.052 - 1.9873 - 61 mat

Collect the cubic spline components for C(j,u) abcd i 1 X rows 1 - range hs i el s i 1 + el s i el - : hy i el Y i 1 + el Y i el - : hx i el X i 1 + el X i el - : 31 line for i 1 X rows 1 - range a i el hs i el 6 hx i el * / : b i el s i el 2 / : c i el hy i el hx i el / hx i el 6 / s i 1 + el 2 s i el * + (* - (: d i el Y i el : 41 line for a b c d augment 31 line :

abcd 0.7753 0 0.0692 2.358 - 0.9507 - 1.2095 0.6982 2.213 - 0.8943 1.6425 - 0.2652 1.256 - 0.4465 - 1.0404 0.3369 - 1.739 - 0.0997 0.2991 - 0.4044 1.482 - 54 mat

Recover segments coeff's [a,b,c,d]

a abcd 1 col :

c abcd 3 col :

b abcd 2 col :

d abcd 4 col :

j u C i j : a i el u X i el - (3^* b i el u X i el - (2^* + c i el u X i el - (* + d i el + 21 line :

plotXY 1 x C 2 x C 3 x C 4 x C 5 x C XY 61 sys :

plotXY

About Hermite cubic . . .

is of different construction, segments junctions are discontinuous.

Will fit better some Gaussian. Also called "Draftsman spline".

The Mathcad Odesolve wrapper is essentially built around lspline

[linear spline] and can't be extrapolated. In other types of works,

be careful if you extrapolate.

Both Hermite, l_p_c_splines can be constrained at end points,

just manipulate the 1rst and last 's' coefficients,

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Global pts plot

Sy . :

size 14 :

clr black :

ptsSpec very fast plot points k 1 : i k X rows range data i el Sy size clr 13 mat : k k 1 + : 21 line for Unest Points v data 1 el : j 2 data rows range v v data j el stack : for v 71 line :

On large number of points, plotG(vx,vy,a,b,c) is prohibitive of computation time. "ptsSpec" is about 12 times faster.

Data X Y ptsSpec augment :

plot 1 x C 2 x C 3 x C 4 x C 5 x C Data 61 sys :

Cubic spline segments through data points

plot

2 2.5 - C 1.680375 -

piecewise integration compatible Σ X XY 1 col : i 2 X rows range vΣ i el i 1 - x C x X i 1 - el X i el int : for vΣ sum 31 line :

Σ 7.3752 -