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Part 1. get the 's' spline coefficients

t0 1 time :

L H N xd range discretise U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 51 line :

L 0 : H 10 : Num 100 : 31 line

U L H Num xd :

x φ 11 x 5 - (2^+ / :

Data i 1 Num 1 + range vy i el U i el φ eval : for U vy augment 21 line :

Data

===== generalise from here =====

X Data 1 col : Y Data 2 col : n Data rows : 31 line

N n 1 - : r 2 N range : 21 line

Prepare the cubic splines

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

Initialise the blank diagonal matrix

100000000000000000000000155 mat

M 11 el 1 :

M n n el 1 :

Fill-in the respective l_p_c splines tridiagonal matrices

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

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

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 - (/ :

M M :

s M 1 -^br * :

CWD C:\SmathFile CurrentDirectory :

Here, we collect the 'cspline'. Disable the 'cspline' tridiagonal to collect 'pspline' ... disable 'pspline' tridiagonal to collect 'lspline.

Spool to file the spline coefficcients s CWD spline wfile 1

s CWD spline rfile :

Part 2. The complimentary cumulative integrate

Prepare the pseudo-integrand s X Y fintegrate i 1 N range h i el X i 1 + el X i el - : f i 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 :

Σ s X Y fintegrate :

Σ Σ :

v X 1 Σ rows 11 submatrix :

view the pseudo-integrand

v Σ augment

Export the cumulative integral Σcumul n X rows 1 - : Sum 1 el 0 : Sum n 1 + el 0 : i 1 n range Sum i 1 + el Sum i el Σ i el + eval : for X Sum augment 51 line :

x MyInt x φ x int maple : 5 - x + atan

l_p_c spline cumulative integrate

Σcumul 0 2 1 sys

Symbolic cumulative Integral

0 x ≤ ( x H ≤ ( & ( x MyInt 0 MyInt - Maple Symbolic if 0 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 Dsymbolic x φ x 2 diff :

x D s X Y x dspline :

3 D 0.16

x plotD L x ≤ (x H ≤ (& (x D out spline if :

does NOT sub-derive

x plotSym L x ≤ (x H ≤ (& (x Dsymbolic out spline if :

x D2 x D x diff :

The scalar dspline

x plotD 0 2 1 sys

2nd derivative of [lpc] spline

X s augment 0 2 1 sys

Symbolic derivative

x plotSym 0 2 1 sys

Observe 2nd derivative

Project Note: In this demo, we have used a function. In real life you will just have a data set and no 2nd symbolic unless you have fitted by some method. In the case of "just a data set", both derivatives will be 0K. Either l_p_c splines in this document are "classical", i.e: same as from reputed CAS [Mathcad, Matlab ...]. They are: "analytical up to the 2nd order derivative".

Part 4. The [lpc] spline interpolation

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 Spline s X Y x interpspline :

x plot L x ≤ (x H ≤ (& (x Spline out spline if :

Sources and credits: 1. Mathsoft for the tridiagonal systems. 2. Robert Adair for the algos: ...InterpSpline(s,X,Y,t) ...dspline(s,X,Y,t) ...fintegrate(s,vx,vy) 3. Smath adapted: Jean Giraud

"Proofing" the cubic spline project

x plot

jmGiraud: 2016_02_18

Revisit associated infinitesimal analysis:

... from infinitesimal analysis of the scalar dspline(s,X,Y,x) ... we get the 2nd & 3rd derivative [scalar canvas, interpolation].

x f s X Y x dspline :

h 0.1 :

x h f D1 x 2 h * - f x 2 h * + f - 4 h * / - :

x h f D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / :

x plotf L x ≤ (x H ≤ (& (x f out spline if :

x plotD1 L x ≤ (x H ≤ (& (x h f D1 out spline if :

x plotD2 L x ≤ (x H ≤ (& (x h f D2 out spline if :

The scalar dspline

x plotf 0 2 1 sys

The scalar 2nd derivative

x plotD1 0 2 1 sys

The scalar 3rd derivative

x plotD2 0 2 1 sys

1 time t0 - 114 s *

The long cubic spline [lpc], scalar interpolation. s X Y t InterpSpline 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 + 41 line :