<p> Mathcad 8 Pro. code conversion </p>

adff50dc-8cc4-4de8-9883-1a00493204d1 57 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

The 3 types of "cubic spline"

3 appVersion 0.98.6179 ≡

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

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 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 :

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 v data 1 el : j 2 data rows range v v data j el stack : for v 51 line :

support X Y ptsSpec augment :

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

2 2.5 - C 1.680375 -

Cubic spline segments through data points

plot

support 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 65 mat

Demo: collect the 'j' segments [if there are few]. 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 :

x segments x X 1 el ≥ (x X X rows el ≤ (& x Spline Not interpolated if :

"Proofing" the cubic spline project

x segments 4 x C 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 3 1 sys

We want a more general tool for lot more works

<p> Mathcad 8 Pro. code conversion </p>

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 Smath X Y x ainterp :

collect/simplify plot x cubic X 1 el x ≤ (x X X rows el ≤ (& (s X Y x InterpSpline Don't interpolate if :

"Proofing" the cubic spline project

x cubic X 1 el Y 1 el . 14 black X 2 el Y 2 el . 14 black X 3 el Y 3 el . 14 black X 4 el Y 4 el . 14 black X 5 el Y 5 el . 14 black X 6 el Y 6 el . 14 black 6 5 mat 2 1 sys

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 41 line :

L 3.52 - : H 1 : Npts 100 : 13 mat U L H Npts xd : 21 mat

Ω i 1 U rows range vy i el U i el cubic eval : for U vy augment 21 line :

Discrete, user segment, Data

Ω support 2 1 sys

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 Deriv L x ≤ (x H ≤ (& (s X Y x dspline out spline if :

1.5 - Deriv 0.706575 -

Observe 2nd derivative

The scalar dspline 1rst Derivative

x Deriv 0 2 1 sys

2nd derivative of [lpc] spline

X s augment 0 2 1 sys

t0 1 time :

data vx L H 100 xd : i 1 vx rows range vy i el vx i el cubic eval : for vx vy augment 31 line :

'x' is NOT scalar wrt Simpson integrator
but recognizes Smath(x):=cinterp(X,Y,x) x cubic vectorize x L 0 int

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

Discrete cumulative integration

data data Σ support 3 1 sys

1 time t0 - 2.215 s *

x Smath X Y x cinterp :

data Σ data Σ rows row 1 7.3752 - 12 mat

x Smath x L 1 int 7.3814 -

t0 1 time :

Σ vx L H 100 xd : i 1 vx rows range Cumul i el x Smath x vx 1 el vx i el int eval : for vx Cumul augment 31 line :

1 time t0 - 12.121 s *

Algo style Memo formula of incremental analysis of this demo example h 0.0001 : (increment ≡ x h f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x h f D1 x 2 h * - f x 2 h * + f - 4 h * / - : (1rst Derivative ≡ 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^* / : (2nd Derivative ≡ 41 line

x f x Smath :

x c 1 L x ≤ (x H ≤ (& cases :

x h f D1 x c * x h f D2 x c * 2 1 sys

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 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 :

t0 1 time :

Σ s X Y fintegrate :

Σ Σ :

v X 1 Σ rows range 11 range el :

view the pseudo-integrand

v Σ augment

Cumulative integration @ data points Σcumul n X rows 1 - : Sum 1 el 0 : i 1 n range Sum i 1 + el Sum i el Σ i el + eval : for X Sum augment 41 line :

Σ 1.2026 - 1.6984 - 1.4473 - 1.6723 - 1.3546 - 51 mat

Σ sum 7.3752 -

Σcumul 3.52 - 03 - 1.2026 - 2 - 2.901 - 1 - 4.3484 - 0 6.0206 - 1 7.3752 - 62 mat

1 time t0 - 0.093 s *

FD => 1.2 sec

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

Σsegments Σcumul + 12 red plot :

spline cumulative @ data points

Σcumul Σsegments 2 1 sys

Reconciliation . . .

1. Simpson integrator may be inaccurate to useless

. . . practically slow, depends upon acc.

2. Finite differences quite fast, very accurate

considered yard-stick in other types of applications.

! insensitive ! to reasonable noisy data sets.

3. fintegrate(s,X,Y) is exact, hyper-fast

. . . interesting cumulative @ data points

]]>

The complimentary spline arc length

arc vx L H 100 xd : i 1 vx rows range vy i el 1 s X Y vx i el dspline (2^+ sqrt eval : for vx vy augment 31 line :

user 50 :

Xuser arc 1 col user el : 1.3052 -

t0 1 time :

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

Discrete cumulative integration

Ω arc Σ Xuser 10 Xuser 10 - 2 2 mat 3 1 sys

1 time t0 - 1.8 s *

arc Σ arc Σ rows row arc Σ user row 21 mat 1 5.1666 12 mat 1.3052 - 2.7309 12 mat 21 mat

u v axes u v / - :

Ω arc Σ Xuser 10 Xuser 10 - 2 2 mat axes XY 5 1 sys

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