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The Derivative & Integral cspline project

x f 11 x 2^+ / :

f L H N n Tabular dx H L - N / : i 1 N range U i el L dx - (dx i * + : for i 1 U rows range vx i el U i el : vy i el U i el (f n round eval : 21 line for vx vy augment 41 line :

data f 5 - 52009 Tabular :

X data 1 col :

Y data 2 col :

f data 2 col :

h X X length el X 1 el - data rows 1 - / : 120 /

Lo 5 - 15 - 1 - 22 mat :

Hi 5151 - 22 mat :

H F.B. HILDEBRAND (3.11.9) ... derivative @ tabular ordinates i 5 data rows 5 - range D i el 1840 h * / (3 f i 4 - el * 32 f i 3 - el * - 168 f i 2 - el * + 672 f i 1 - el * - 672 f i 1 + el * + 168 f i 2 + el * - 32 f i 3 + el * + 3 f i 4 + el * - (* : for X 1 D rows 11 submatrix D augment 31 line :

Derivative at tabular points . For the plot, X data are added. If you don't have equally spaced points, interpolate/discretise [L,H,N]

H data 2 1 sys

x q 11 x 2^+ / : points generatrix ≡

x q x q x diff Lo Hi 4 1 sys

Lo, Hi plot on pure function but not over "freaked" fnct.

u Φ 11 x 2^+ / :

S(x) is the native Smath algo x S X Y x cinterp eval :

x Plot u Φ u x - x int maple : 2 x * 1 x 2^+ /

u F x S :

<= assign S(x) as the integrand

F(u) is the cspline that will run 
the integral over the plot canvas. z Int u F u x - x int :

x Plot

back conversion x Int z Int :

Integral curve of splined data

5 - x ≤ ( x 5 ≤ ( & x Int OUT of data set range if

Explain the triplet integral curve: S(x) is simply a dummy algo. It is assigned dummy again as the integrand. Int(z) is the 3rd dummy runner algo. At this point, nothing is calculated. Smath plots only over 'x', Int(z) is converted back in the canvas 'x'. Things will calculate and plot once assigned in the canvas which by default runs over unknown mesh. The plotting canvas is a working pusher, i.e: a running scalar. Each time it wants to go one more step, the all things go back to evaluate S(x). Each calculated point is displayed and accumulated in the plot canvas. It does in succession until the canvas is filled over the specified range and unknown mesh.

1.23456789 Int 0.978202

The Cumulative Integration is the sum of the finite areas under the Integral Curve. It always starts at 0 value. The proposal suits raw data of non equally spaced points.

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 :

Cumulative integration of data

data data Σ 2 1 sys

IntegralCurve Export the Integral x X : k 1 : i 1 x rows range Sum i el x i el Int eval : k k 1 + : 31 line for x Sum augment 51 line :

IntegralCurve

x f x S :

h 120 /

1rst derivative  applied on interpolating functions x h D1 x 2 h * + f - x 2 h * - f + (- 12 h * / :

2nd derivative applied on interpolating functions x h D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / :

x h D3 x 3 h * - f 8 x 2 h * - f * - 13 x h - f * + 13 x h + f * - 8 x 2 h * + f * + x 3 h * + f - 8 h 3^* / :

x h D4 x 3 h * - f - 12 x 2 h * - f * + 39 x h - f * - 56 x f * + 39 x h + f * - 12 x 2 h * + f * + x 3 h * + f - 6 h 4^* / :

5 - x ≤ ( x 5 ≤ ( & x h D1 OUT of data set range if 5 - x ≤ ( x 5 ≤ ( & x h D2 OUT of data set range if 5 - x ≤ ( x 5 ≤ ( & x h D3 OUT of data set range if

Data sets that exhibit noise on D2, D3 indicate too low true decimal places. Applying a bit of ksmooth may help, if NOT: nothing to do => attempt fitting a model function. NOT an easy task to join fitted segments !

clean Export the clean D3 x X : h X X length el X 1 el - X rows 1 - / : i 1 x rows range Deriv i el x i el h D3 eval : for x Deriv augment 51 line :

clean Lo Hi 3 1 sys

Thumb rule: ndecimals sampled data derivative order 'n' ~ ndecimals^2

Apply some reverse calculus on our original pretention

u f 2 x * 1 x 2^+ / : maple integral curve ≡

x Plot u f u 0 x int maple : 2 x 2^* 1 x 2^+ /

1012^Plot 2

x Plot - 1012^Plot + 2 / 11 x 2^+ /

order 3 :

x Plot - 10 12 ^ Plot + 2 / 2 1 sys 1 1 x 2 ^ + / 1 1 x 2 ^ + / ( x order diff

11 x 2^+ /

2 x * 1 x 2^+ /

11 x 2^+ / x int maple x atan

2 x * 1 x 2^+ / x int maple 1 x 2^+ ln

1 1 x 2 ^ + / x atan 2 1 sys 2 x * 1 x 2 ^ + / 1 x 2 ^ + ln 2 1 sys