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Spline cumulative integral

From the l_p_c cubic spline procedure, you can export the cumulative integral in two simple steps of construction. The procedure suits not equally spaced data.

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 21 line 41 line :

L 0 :

H 10 :

Num 100 :

U L H Num xd :

n U rows :

n 101

x φ 11 x 5 - (2^+ / :

Data i 1 n range vx i el U i el : vy i el U i el φ : 21 line for vx vy augment 21 line :

Data

X Data 1 col :

Y Data 2 col :

N n 1 - :

N 100

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

Prep.1 Install the tridiagonal matrix

M 11 el 1 :

M N 1 + N 1 + el 1 :

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 - (/ :

s M 1 -^br * 11 line :

s 0.0083754 0.0090863 0.0097973 0.0105981 0.0114754 0.012445 0.0135167 0.0147039 0.0160212 0.0174857 0.0191171 0.020938 0.0229745 0.025257 0.0278205 0.030706 0.033961 0.0376411 0.041811 0.0465467 0.0519368 0.0580854 0.065114 0.0731648 0.0824036 0.0930232 0.1052462 0.1193271 0.1355536 0.1542437 0.1757375 0.2003802 0.228488 0.2602891 0.295825 0.3347893 0.3762773 0.4184078 0.4577791 0.4887368 0.5024878 0.4862373 0.422796 0.2915381 0.0720532 0.2481284 - 0.6616898 - 1.1291553 - 1.574664 - 1.9002211 - 2.0201865 - 1.9002211 - 1.574664 - 1.1291553 - 0.6616898 - 0.2481284 - 0.0720532 0.2915381 0.422796 0.4862373 0.5024878 0.4887368 0.4577791 0.4184078 0.3762773 0.3347893 0.295825 0.2602891 0.228488 0.2003802 0.1757375 0.1542437 0.1355536 0.1193271 0.1052462 0.0930232 0.0824036 0.0731648 0.065114 0.0580854 0.0519368 0.0465467 0.041811 0.0376411 0.033961 0.030706 0.0278205 0.025257 0.0229745 0.020938 0.0191171 0.0174857 0.0160212 0.0147039 0.0135167 0.012445 0.0114754 0.0105981 0.0097973 0.0090863 0.0083754 1011 mat :

Part 2. construct the cumulative integral

On this line, insert/isolate s:= ... s [spline coeffs] that you have exploded =>

On this line or above, plug the previously exploded 's'

vx X :

vy Y :

s rows 101

Prepare the pseudo-integrand s vx vy fintegrate i 1 n 1 - range h i el vx i 1 + el vx i el - : f i el h i el 2 / vy i el vy i 1 + el + (* h i el (3^24 / s i el s i 1 + el + (* - : 21 line for f 21 line :

f s vx vy fintegrate :

v vx 1 f rows 11 submatrix :

f rows 100

v rows 100

view the pseudo-integrand

v f augment

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

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

sl_p_c spline cumulative integral

Σcumul 0 2 1 sys

Symbolic cumulative Integral

x 0 ≥ ( x 10 ≤ ( & x MyInt 0 MyInt - Maple Symbolic if 0 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 :

x D s X Y x dspline :

x D2 x D x diff :

The 2nd derivative of spline, i.e: derivative of "dspline" is granular wrt the Smath derivative operator and will NOT plot/export. In lieu, collect the 's' coefficients. Not smooth but visual.

D2nd i 1 n range U i el U i el : D2nd i el U i el D2 eval : 21 line for U D2nd augment 21 line :

x Dsymbolic x φ x 2 diff :

The scalar dspline

x 0 ≥ ( x 10 ≤ ( & x D dspline done if 0 2 1 sys

2nd drivative of c-spline

X s augment 0 2 1 sys x 0 ≥ ( x 10 ≤ ( & x Dsymbolic dspline done if 0 2 1 sys

1 time t0 - 6.9 sec *