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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 constanr vector 'b' 2. Install the basis 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

M 11 el 1 :

M N 1 + N 1 + el 1 :

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 4.0766 1.6415 3.0412 - 1.8833 0.052 - 1.9873 - 61 mat

Construct the general spline cubic polynomial

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

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

Collect the cubic spline components

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 1.0172 0.2501 - 2.358 - 0.868 - 1.0172 0.8078 2.213 - 0.8656 1.5868 - 0.2382 1.256 - 0.4144 - 1.01 0.3386 - 1.739 - 0 0.2332 - 0.4382 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 :

Global pts plot

Sy . :

size 14 :

clr black :

ptsSpec k 1 : i k X rows range data i el Sy size clr 13 mat : k k 1 + : 21 line for 21 line Unest P v data 1 el : j 2 data rows range v v data j el stack : for v 31 line 31 line :

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

u Spline j local assignment n X rows : u X 1 el ≤ n 1 : u X n el ≥ n n 1 - : i 1 n range u X i el ≤ break return 'j' < (i-1) if for if if i 1 - (41 line : j u C 21 line :

"Proofing" the cubic spline project

x X 1 el ≥ ( x X X rows el ≤ ( & x Spline Job done ! if 3.52 - 2.358 - . 14 black 1 5 mat 3 - 2.213 - . 14 black 1 5 mat 2 - 1.256 - . 14 black 1 5 mat 1 - 1.739 - + 20 red 1 5 mat 0 1.482 - . 14 black 1 5 mat 1 1.277 - . 14 black 1 5 mat 4 x C 8 1 sys

Data pairs can be added individually but NOT globally. The bad is longer construction, the good is individual specification(s). You can collect all that stuff to cut the mustard and clean your work for publishing. Here, we highlight a selected cubic segment. You can add more as you wish !

Global collection of points spec NOT accepted

The individual plot points specs is fast from copy/paste each row at time from "ptsSpec". In other words: by "unesting" manually each row.

ptsSpec k 1 : i k X rows range data i el X i el Y i el Sy size clr 15 mat : k k 1 + : 21 line for 21 line data 21 line :

ptsSpec 3.52 - 2.358 - . 14 black 15 mat 3 - 2.213 - . 14 black 15 mat 2 - 1.256 - . 14 black 15 mat 1 - 1.739 - . 14 black 15 mat 0 1.482 - . 14 black 15 mat 1 1.277 - . 14 black 15 mat 61 mat

n_row 3 :

explode individual nest

ptsSpec n_row 1 el 2 - 1.256 - . 14 black 15 mat

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 :

x MySpline s X Y x InterpSpline :

"Proofing" the cubic spline project

x 4 - ≥ ( x 1.75 ≤ ( & x MySpline Job done ! if 3.52 - 2.358 - . 14 black 1 5 mat 3 - 2.213 - . 14 black 1 5 mat 2 - 1.256 - . 14 black 1 5 mat 1 - 1.739 - + 20 red 1 5 mat 0 1.482 - . 14 black 1 5 mat 1 1.277 - . 14 black 1 5 mat 3 x C 8 1 sys

Sanity i 1 X rows range vy i el X i el MySpline : vx i el X i el : 21 line for vx vy augment 21 line :

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

0.123456789 MySpline 1.4315 -

Discretise for export

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 21 line 31 line :

L 4 - :

H 1.25 :

Num 200 :

U L H Num xd :

Ω i 1 U rows range vy i el U i el MySpline : vx i el U i el : 21 line for vx vy augment 21 line :

x D x MySpline x diff :

Discrete, data, user segment

Ω Data 3 x C 5 x C 4 1 sys

Granularity in the range

x X 1 el ≥ ( x X 2 el ≤ ( & x D granularity in the range if

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - range deriv i el 0.5 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 - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X deriv augment 101 line :

Ω Deriv

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 :

t deriv s X Y t dspline :

The scalar dspline

x 4 - ≥ ( x 1.75 ≤ ( & x deriv dspline done if 0 2 1 sys

Dspline i 1 U rows range vy i el U i el deriv : vx i el U i el : 21 line for vx vy augment 21 line :

Dspline 0 2 1 sys

jmGiraud: 2015/09/24