Prepare the Tridiagonal Matrices

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. . . ► l_p_c cubic spline coefficients

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Spline Polyline_10 [l,p,c]

Part 1. l_p_c cubic scalar splines

4 appVersion 0.98.6179.21440 ≡

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: l, p, c] two steps of preparation are needed [collapsed area]: 0. Install the constant vector 'br' 1. 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 - : 5

r 2 N range :

Prepare the Tridiagonal Matrices

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 - / (- eval : br N 1 + el 0 : 21 line for

br 0 0.678153846153846 1.44 - 0.74 0.052 - 061 mat

Prep.1 Install the tridiagonal matrix

M 11 el 1 :

M N 1 + N 1 + el 1 :

The basis tridiagonal matrix [l, p, c] M 10000000000000000000000000000000000166 mat

l, p, c tridiagonal construction

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

slin M 1 -^br * eval :

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

slin 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 - :

sp M 1 -^br * eval :

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

sp 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 - (/ :

sc M 1 -^br * eval :

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

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

linear slin 0 2.419 3.285 - 2.0808 0.5982 - 061 mat

parabolic sp 2.0344 2.0344 3.1736 - 2.02 0.4664 - 0.4664 - 61 mat

cubic sc 4.0766 1.6415 3.0412 - 1.8833 0.052 - 1.9873 - 61 mat

◄ Sanity Mathcad 11

In 's' , plug the the selected type of spline [slin, sp,sc]
'slin & sp' join cubic 2 points entrance and along and departutre.
'sc' joins cubic 3 points entrance and along and departure. s slin :

Type linear s slin ≡ parabolic s sp ≡ cubic cases :

Part 2. Classroom journey cubic splines

abcd

Collect the cubic 's' 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 0.6981 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 :

ptsSpec

ptsSpec k 1 : i k X rows range data i el Sym size clr 13 mat : k k 1 + : 21 line for v data 1 el : j 2 data rows range v v data j el stack : for v 51 line :

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

Sym . :

size 14 :

clr black :

support X Y ptsSpec augment :

J 4 :

Type linear

x λ 12 - x ≤ (x 2 ≤ (& cases :

Cubic spline segments through data points

plot

cubic segment(s) observer 'J' "cubic" C(4,x)= C(5,x)

5 x C x λ * J x C support 3 1 sys Mathcad 8 Pro. code conversion

s X Y t InterpSpline Mathcad 8 Pro. code conversion t X 1 el ≤ idx 1221 mat : t X N el ≥ idx N N 1 + 21 mat : i 1 : X i el t < i i 1 + : while idx i 1 - i 21 mat : 31 line if if hx X idx 2 el el X idx 1 el el - (: hy Y idx 2 el el Y idx 1 el el - (: hs s idx 2 el el s idx 1 el el - (: 31 line a hs 6 hx * / : b s idx 1 el el 2 / : c hy hx / hx 6 / s idx 2 el el 2 s idx 1 el el * + (* - (: d Y idx 1 el el : 41 line a t X idx 1 el el - (3^* b t X idx 1 el el - (2^* + c t X idx 1 el el - (* + d + 51 line :

◄ 's' from your collection above

recast the 'interpolation' code x spline s X Y x InterpSpline :

interpolate within 'x range' 1.5 - spline 1.4222 -

x smath X Y x cinterp :

Type linear

1.5 - smath 1.4234 -

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

support XY . 14 black plot :

x λ 14 - x ≤ (x 2 ≤ (& cases :

x f x spline :

3.52 - 2.358 - . 14 black 3 - 2.213 - . 14 black 2 - 1.256 - . 14 black 1 - 1.739 - . 14 black 0 1.482 - . 14 black 1 1.277 - . 14 black 6 5 mat x f 2 1 sys

h 0.0001 :

x h f D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / :

x D1 x h f D1 :

1.5 - D1 0.7066 -

x D1 x λ * x f x λ * 2 1 sys

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 3.5 - x ≤ (x 1.5 ≤ (& cases : 21 sys

x D x c *

rootsDichotomy local command line style x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 21 line 31 line :

x f x D :

This roots session is demo for "linear" ... else exercise.

on failure , increase TOL TOL 1012 -^:

r 1 - 0.75 - TOL D 2 - 1.75 - TOL D 21 mat : 0.816389432080 - 1.913105383620 - 21 mat

Part 4. The complimentary fintegrate

Prepare the pseudo-integrand ... 'N' from initial  spline setup s X Y fintegrate i 1 N range h i el X i 1 + el X i el - : f i 1 + el h i el 2 / Y i el Y i 1 + el + (* h i el (3^24 / s i el s i 1 + el + (* - : 21 line for f 21 line :

s X Y fintegrate sum 7.3752 -

Part 5. Scalar cumulative integration

Σ X s X Y fintegrate augment :

x cumul s Σ 1 col Σ 2 col x InterpSpline :

in-situ command line u 1 - :

x cumul x c * u u cumul + 20 red 1 5 mat 2 1 sys

Σ 3.52 - 03 - 1.2026 - 2 - 1.6984 - 1 - 1.4473 - 0 1.6723 - 1 1.3546 - 62 mat

u u cumul 12 mat 1 - 1.4473 - 12 mat