45ae93f9-c19c-45ea-9ed4-23d4e20f6087 38 6 5 auto

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

u v Kernel i 1 u rows range j 1 u rows v rows + range H i j el 0 : for for i 1 u rows range jj 1 v rows range H i jj i + el v jj el : for for Kernel H : 31 line :

x 0.1 0.2 0.3 0.4 41 mat :

h 1.123456789 2.2 3.3 4.4 41 mat :

K x h Kernel :

K 0 1.123457 2.2 3.3 4.4 00000 1.123457 2.2 3.3 4.4 00000 1.123457 2.2 3.3 4.4 00000 1.123457 2.2 3.3 4.4 48 mat

Matrix/Vector Convolution

K 1.123456 2.2 3.3 4.4 00000 1.123456 2.2 3.3 4.4 00000 1.123456 2.2 3.3 4.4 00000 1.123456 2.2 3.3 4.4 048 mat :

x 0.1 0.2 0.3 0.4 41 mat :

Convolve matrix => vector Convol C x transpose K * (transpose eval : i 1 C rows range v i el i : 11 line for v C augment 31 line :

Transit result XY Convol :

Convol tabulates but does not plot

i 1 Convol rows range p i el . : for

K x transpose * freaks the syntax rule ≡

data XY 1 col XY 2 col p augment :

data XY 2 1 sys

Optimiz: numeric, Decimal Convol 1 0.1123 2 0.4447 3 1.107 4 2.2094 5 2.75 6 2.64 7 1.76 8082 mat

u v Convol smae as Mathcad built-in i 1 u rows range j 1 u rows v rows + range H i j el 0 : for for i 1 u rows range jj 1 v rows range H i jj i + 1 - el v jj el : for for u transpose H * (transpose 41 line :

x h Convol 0.112346 0.444691 1.107037 2.209383 2.75 2.64 1.76 081 mat

For this convoved vector, it is tempting to fit a function. A 5th degree polyfit is OK. The "user scale" is 0.5 in this polyfit demo. We can shift the fit, unshifted fir is better.

C 0.112346 0.444691 1.107037 2.209383 2.75 2.64 1.76 081 mat :

scale 0.5 :

shift 0 :

XY i 1 C rows range X i el i scale / : for X C augment 21 line :

x XY 1 col shift - :

y XY 2 col :

degree 5 :

XY 2 0.112346 4 0.444691 6 1.107037 8 2.209383 10 2.75 12 2.64 14 1.76 16082 mat

i 1 y length range j 1 degree range M i j el x i el (j^: for for 11 line

Cholesky solver k M transpose M * (1 -^M transpose * y * :

if not in that order: f..k it

y M transpose * M transpose M * (1 -^*

not even commutative

k 0.088525 0.060087 - 0.024097 0.002169 - 5.474234 105 -^* 51 mat

u f k i el u i^* (i 1 k length sum :

i 1 XY rows range p i el . : for

data XY 1 col shift - XY 2 col p augment :

User scaled polyfit. Note that some extrapolation is possible.

data x f 2 1 sys

Simple convolution

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

Simple convolution:CONVOL(x,h,n) is often confusing in CAS, with double convolution from above Convol(x,h). Let disambiguate by plotting both styles below.

x h n CONVOL simple convol prepare m n : x k 1 + el h m k - 1 + el * (k 0 m sum 31 line :

u v C simple convol construct curiosiy: N is wild i 0 N range C i 1 + el u v i CONVOL : for v C augment 41 line :

N 15 :

convol vectors n 0 N range x n 1 + el n 2 / : h n 1 + el n : 21 line for 21 line

Ω Double convolution Ω x h Convol : i 1 Ω length range u i el i 1 - : for u Ω augment 41 line :

C x h C :

C C :

C 00102 0.5 32455106 17.5 72884296010 82.5 11110121431318214 227.5 15280162 mat

x 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 161 mat

h 0123456789101112131415161 mat

Ω Ω :

a b c spec . 12 red 13 mat :

ptsΩ Ω a b c spec pts :

a b c spec o 7 black 13 mat :

ptsC C a b c spec pts :

Ω => Double convolution as smooth trace ptsΩ => Double convolution plotted points ptsC => Simple convolution plotted points

Ω ptsΩ ptsC 3 1 sys