<p> Join the segments ... myData spline type [l_p

82930fbf-7acb-44c4-b396-d6a9d167a108 33 15 12 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

lspline 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

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

pspline 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

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 - (/ :

cspline 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

data n Mround i 1 data rows range j 1 data cols range m i j el data i j el n round : for for m 21 line :

s lspline pspline cspline augment :

s s 3 Mround :

s 0 2.034 7.166 2.419 2.034 1.062 3.285 - 3.174 - 2.887 - 2.081 2.02 1.845 0.598 - 0.466 - 0.052 - 0 0.466 - 1.949 - 63 mat

spline type type 2 :

1 linear end points ≡ 2 parabolic end points ≡ 3 cubic end points ≡ 31 mat

s s type col :

Construct the general spline cubic polynomial

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 1.017 0.25 - 2.358 - 0.868 - 1.017 0.808 2.213 - 0.8657 1.587 - 0.2383 1.256 - 0.4143 - 1.01 0.3387 - 1.739 - 0 0.233 - 0.438 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 :

x j sanity a j el x X j el - (3^* b j el x X j el - (2^* + c j el x X j el - (* + d j el + :

<p> Join the segments ... myData spline type [l_p_c] </p>

The while loop form of the spline interpolation function s X Y t interpspline Find the knots that bracket 't' i0 1 : j X rows : j i0 - 1 > i i0 j + 2 / Floor : X i el t > j i : i0 i : if 21 line while Evaluate the spline based on those knots h X j el X i0 el - : a X j el t - h / : b t X i0 el - h / : a Y i0 el * b Y j el * + h 2^6 / a a 2^1 - (* s i0 el * b b 2^1 - (* s j el * + (* + 91 line :

X_Y rejects the scalar plot x mySpline s X Y x interpspline :

myData x 3.75 - 1.5 3.7 - range : j 1 x rows range vy j el x j el mySpline eval : for x vy augment 31 line :

myData myData :

myData

plotXY 1 x C 2 x C 3 x C 4 x C 5 x C XY myData 71 sys :

plotXY

type 2

Observe . . .

type 2 LHS tracks segment 2

type 3 LHS tracks segment 1

]]> j x j sanity

j x C 25973000 / x 2 - (- (3^* 15871000 / - (x 2 - (- (2^* + 143600 / x 2 - (- (* + 157125 / - +

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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Sy . :

size 14 :

clr black :

Global pts plot

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 1.66325 -

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 7.3646 -

Part 3. The complimentary scalar dspline

s X Y t dspline t X 2 el ≤ k 1 : t X X rows 1 - el ≥ k X rows 1 - : k 1 : t X k 1 + el > k k 1 + : while 21 line if if c Y k 1 + el Y k el - X k 1 + el X k el - / s k 1 + el 2 s k el * + (X k 1 + el X k el - 6 / (* - : s k 1 + el s k el - X k 1 + el X k el - / (t X k el - (2^2 / * s k el t X k el - (* + c + 31 line eval :

x D s X Y x dspline :

x c 1 3.75 - x ≤ (x 1.5 ≤ (& cases :

x D x c *