<content> Library </content>

f8651a64-c3ed-4660-b007-5d86a62898c3 29 4 5 false false 0 decimal

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

<content> Library </content> <content> Utils </content>

fs# x# feval fs# num2str (x# num2str) concat str2num 11 line :

fs# x# y# feval fs# num2str (x# num2str, y# num2str) concat str2num 11 line :

P cyl2xyz P 1 el P 2 el cos * P 1 el P 2 el sin * P 3 el 31 mat :

P ρ φ z 31 mat ≡

x# IsMatrix x# length 0 > eval 0 try 11 line 10 if 11 line :

a b 12 mat GetType unknown

a b 12 mat IsMatrix 1

z# ReIm z# Re z# Im augment :

z# Arg z# 0 ≡ 0 z# arg if 11 line :

M# iFind M M# eval : n 1 M length range : 12 mat M 0 ≡ (vectorize n el n augment eval 11 findrows 2 col eval 0 try 21 line :

rnd 109 -^109^random * 11 line :

a b rnd a b a - (rnd * + 11 line :

a b n rnd X 0 : r 1 n range : 12 mat X r el a b rnd : 21 line :

a noise a - a rnd 11 line :

MCols(M) returns the columns of M as a row vector. M MCols n M cols : R 1 n matrix : c 1 n range : 13 mat R c el M c col : eval 21 line :

<content> pUtils </content>

fs# x# pFeval eval(try(fs# num2str (x# num2str),'NaN)) concat str2num 11 line :

fs# x# y# pFeval eval(try(fs# num2str (x# num2str, y# num2str),'NaN)) concat str2num 11 line :

range utility a b n pR a b a - n / 0 n range * + eval 11 line :

M p# pBox p p# p# p# stack eval : M cols 2 ≡ X Y 12 mat M 1 col M 2 col 12 mat eval : B X min X max Y min Y max 22 mat eval : p 1 el B 1 el abs * - p 1 el B 2 el abs * p 2 el B 3 el abs * - p 2 el B 4 el abs * 22 mat B + eval 31 line X Y Z 13 mat M 1 col M 2 col M 3 col 13 mat eval : B X min X max Y min Y max Z min Z max 32 mat eval : p 1 el B 1 el abs * - p 1 el B 2 el abs * p 2 el B 3 el abs * - p 2 el B 4 el abs * p 3 el B 5 el abs * - p 3 el B 6 el abs * 32 mat B + eval 31 line if 21 line :

P cl ls w cs pLine cs 0 ≡ polyline P 11 mat cl ls w stack eval polygon P 11 mat cl ls w cs stack eval if 11 line :

P C pLine polyline P 11 mat C solid 1 stack eval 11 line :

P C pPolygon polygon P 11 mat C solid 1 C stack eval 11 line :

DS# xo yo te pRK t# u# D_RK# DS# u# 1 el u# 2 el feval eval : xo yo stack eval 0 te 100 D_RK# rkfixed 1101 range 2312 mat el eval 21 line :

x y z pAxis 000 x 000000 y 000000 z 00073 mat :

M γ pAxis M cols 3 ≡ A 0 : k 13 range : 12 mat k m 00013 mat 1.05 M k col max * 3 identity k row * stack γ 13 range 12 range el * : A k el polyline k m 11 mat red green magenta 31 mat k el solid 2 stack eval : 31 line todo if 11 line :

<content> pAdapt </content>

z w v pCAngle a w z - : b v w - : n a abs b abs * eval : 13 mat n 0 ≡ 0 a Re i a Im * - (b * Re n / acos Re eval if 21 line :

fs a b No α p MI pAdapt t 0 : T a b No pR : Z fs T pZGrid : 13 mat d p Z abs vectorize max Z abs vectorize min - abs * eval : iter 1 MI range N T length : A 0 : D 0 : 13 mat r 1 N 2 - (range : A r el Z r el Z r 1 + el Z r 2 + el pCAngle : 12 mat r 1 N 1 - (range : D r el Z r 1 + el Z r el - abs : 12 mat A max α ≤ (D max d ≤ (& eval break N T length : k 0 : t 0 : 13 mat n 0 0.5 N 1 - (* 1 - (range t k k 1 + : el T 2 n * 1 + el : θ Z 2 n * 1 + el Z 2 n * 2 + el Z 2 n * 3 + el pCAngle : θ α > (Z 2 n * 2 + el Z 2 n * 1 + el - abs d > (| t k k 1 + : el 0.5 T 2 n * 1 + el T 2 n * 2 + el + (* : if t k k 1 + : el T 2 n * 2 + el : θ α > (Z 2 n * 2 + el Z 2 n * 3 + el - abs d > (| t k k 1 + : el 0.5 T 2 n * 2 + el T 2 n * 3 + el + (* : if 51 line for N 2 mod 0 ≡ t k k 1 + : el T N 1 - el : θ α > (Z N el Z N 1 - el - abs d > (| t k k 1 + : el 0.5 T N 1 - el T N el + (* : if 21 line if T t T N el stack eval : Z fs T pZGrid : 12 mat 41 line if 41 line for Z 41 line :

fs a b pAdapt fs a b pAdapt 1 el pAdapt 2 el pAdapt 3 el pAdapt 4 el pAdapt 11 line :

pAdapt 2515 ° * 10 % * 814 mat :

<content> pGrid </content>

B# N# pGrid m kg s K A cd mol 17 mat 111111117 mat : B# rows 2 ≡ (B# cols 2 ≡ (& (N# length 2 ≡ (& X# Y# 12 mat B# 1 el B# 2 el N# 1 el pR eval B# 3 el B# 4 el N# 2 el pR eval 12 mat : X# Y# 12 mat B# N# 12 mat eval : if 21 line :

FS# T# pZGrid m kg s K A cd mol 17 mat 111111117 mat : k# 1 T# length range : Z# 0 : M# 0 : M# k# el FS# T# k# el pFeval : 14 mat M# 1 el IsMatrix Z# k# el M# k# el 1 el i M# k# el 2 el * + : M# eval if 31 line :

FS# U V pZGrid m kg s K A cd mol 17 mat 111111117 mat : X Y 12 mat U V pGrid : FS# IsMatrix M FS# eval : r 1 X length range : c 1 Y length range : M 0 : 13 mat M r c el FS# X r el i Y c el * + pFeval : 21 line if 31 line :

FS# X Y pTGrid A pr v A 2 el A 1 el - 2 / A + : v v length el A v length el : eval 21 line : FS# X pr Y pr pGrid 21 line :

FS# X pRGrid m kg s K A cd mol 17 mat 111111117 mat : k 1 X length range : F 0 : 12 mat F k el FS# X k el pFeval : 31 line :

FS# U V pRGrid m kg s K A cd mol 17 mat 111111117 mat : X Y 12 mat U V pGrid : FS# IsMatrix M FS# eval : r 1 X length range : c 1 Y length range : M 0 : 13 mat M r c el FS# X r el Y c el pFeval : 21 line if 31 line :

pGrid(F,U,V) returns a length(U)*lenght(V) matrix with [u,v,F(u,v)] FS# U V pGrid m kg s K A cd mol 17 mat 111111117 mat : FS# IsMatrix FS# eval X Y 12 mat U V pGrid eval : r 1 X length range : c 1 Y length range : M 0 : 13 mat FS# X 1 el Y 1 el pFeval IsMatrix M r c el FS# X r el Y c el pFeval : M r c el X r el Y c el FS# X r el Y c el pFeval 13 mat : if 31 line if 21 line :

<content> pQuiver </content>

Quiver plot of f(x,y)≡[f.1(x,y) f.2(x,y)] FS# U V pQuiver X Y 12 mat U V pGrid eval : F FS# X Y pGrid : 12 mat r 1 X length range : c 1 Y length range : 12 mat n 1 X length Y length * (range : Z 0 : M 0 : XY 0 : 14 mat One 1 pQuiver length 1 matrix + eval : Z n el t F n el norme : (0 ≡ 00 stack eval F n el t / eval if : sc 0.9 Y 2 el Y 1 el - (* Z norme vectorize max / eval : XY r c el X r el i Y c el * + eval : M n el sc Z n el 1 el i Z n el 2 el * + (* pQuiver * XY n el One * + ReIm eval : 81 line :

Arrow -> pQuiver 01 0.8 1 0.8 15 mat 00 0.08 0 0.08 - 15 mat i * + transpose eval :

Arrow -|> pQuiver 01 0.8 1 0.8 0.8 16 mat 00 0.08 0 0.08 - 0.08 16 mat i * + transpose eval :

<content> pView </content>

az el pView α az : ε el : 12 mat ε ε 180 ° * + 360 ° * mod 360 ° * + 360 ° * mod 180 ° * - : ε 90 ° * > ε 180 ° * ε - : α α 180 ° * + : 12 mat ε 90 ° * - < ε 180 ° * - ε - : α α 180 ° * + : 12 mat 0 if if α α 360 ° * mod 360 ° * + 360 ° * mod : α cos ε sin α sin * - ε cos α sin * α sin ε sin α cos * ε cos α cos * - 0 ε cos ε sin 33 mat eval 51 line :

az el pView2 az el pView 13 range 12 range el eval 11 line :

<content> pShow </content>

S# CM γ pShow S S# eval : k 2 S rows range : 12 mat cm S 3 col CM pClr : S2 S γ 13 range 12 range el * eval : L 0 : 13 mat L k el S2 k row S2 k 1 - row stack cm k el pLine eval : pShowAxis L S γ pAxis eval stack eval 11 mat L 11 mat if 41 line :

M# N g GM GS wm ls d pShow M M# eval : nu N 1 el abs : nv N 2 el abs : w wm GM 0 ≠ (* : 14 mat P M g * eval : c 1 P cols range : Z M d col : 13 mat gM gS 12 mat Z GM pClr gS Z GS pClr : 12 mat eval : pShowAxis nL 3 : L M g pAxis eval : 12 mat nL 0 : L 0 : 12 mat if r.np 1 pMesh.np range : r.nv 1 nv 1 - (range : r 0 : S 0 : ns 0 : 15 mat u\0324\ 1 nu range v\0324\ 1 nv range r r 1 + : S ns ns 1 + : el P r r.np + c el gM r el ls w gS r el pLine eval : r r pMesh.np + 1 - : 31 line for r r 1 + u\0324\ nu ≡ (- : L nL nL 1 + : el P r r.nv + c el gM r el ls w 0 pLine eval : r r nv + 2 - : 41 line for nL 0 ≡ 01 matrix L if N 1 el 0 < S reverse S if stack eval 11 mat 71 line :

M N g GM GS pShow M N g GM GS 1 solid 3 pShow 11 line :

<content> pCMap </content>

M# G# pClr M M# eval : G G# eval : n 1 M length range : C 0 : 14 mat G 0 ≡ M length 1 matrix G IsString C n el G : a b m 13 mat M min M max G length 13 mat eval : C n el G M n el a - (m 1 - b a - / (* trunc 1 + el eval : 21 line if if 21 line :

A# pCycleC M# 0 : m# 0 : 12 mat a# A# eval M# m# m# 1 + : el a# : iter 1 pCycleC range M# m# m# 1 + : el ∞ : for 21 line for M# m# m# 1 + : el ∞ : mat2sys.1 eval 31 line :

r g b a rgb hex 0123456789ABCDEF : d 0 : 12 mat c H n 0255 c 0 round 12 mat min 12 mat max : m n 16 mod : 12 mat n m - 16 mod 0 ≡ hex n m - 16 / 1 + 1 substr hex m 1 + 1 substr concat hex m 1 + 1 substr if 21 line : h 255 a * H r H g H b H concat : n h strlen : 12 mat k 1 n range d d hex h k 1 substr findstr 1 el 1 - (16 n k -^* + : for d 232^2 d * 232^1 - ≥ (* - eval 51 line :

C rgb C 1 el C 2 el C 3 el 1 rgb :

C a rgb C 1 el C 2 el C 3 el a rgb :

T n a pCMap C 0 : r 1 n range : 12 mat z r T f 02550 - n 1 - / r 1 - (* + (3 identity z 1 el row * T + a rgb eval 11 line : n 0 ≡ 0 T IsString T Jet ≡ m 0.25 n * 0 round 112 mat max eval : v z v rows 1 matrix : 12 mat x 2551 m range m / * : y 255 m / m 2 / m range (* : o 255 m 1 matrix + : 13 mat R y z o z x o y reverse stack : G y z x o x reverse y z stack : B y o x reverse o z y z stack : m R length n - : C r el R r m + el G r m + el B r m + el a rgb eval : 71 line T GS ≡ R n a pCMap G n a pCMap + B n a pCMap + rgb\0324\ RGB T findstr : (1 - ≡ C r el T : C r el rgb\0324\ r 0 f : if if if C r el T iFind r T f : if if 31 line :

T x pCMap T x 1 pCMap 11 line :

T pCMap T 100 pCMap 11 line :

pCMJet Jet pCMap :

pCMRed R pCMap :

pCMGreen G pCMap :

pCMBlue B pCMap :

<content> pMesh </content>

Main routines

PMesh returns a matrix suitable for 3D plots. F# TF# uo vo pTMeshG F F# F# F# cols col augment eval : TF TF# eval : 12 mat nu F rows 1 - : nv F cols 1 - : M 03 matrix eval : tr 0 : 14 mat np uo length : tp np 1 + : r.np 1 np range : r.nv 1 nv 1 - (range : c 13 range : 15 mat u 1 nu range v 1 nv range r M rows : k r r.np + : tr tr 1 + : 13 mat M k c el F uo k r - el u + vo k r - el v + el c el : v nv ≡ M r tp + c el M r np + c el : M r tp + c el TF tr el c el : if 31 line for r M rows : k r r.nv + : 12 mat M k c el M u 1 - (tp nv * nv + 1 - (* tp v 1 - (* + k r - 1 - (tp * - 1 - c el : 31 line for M 51 line :

PMesh returns a maitrx suitable for 3D plots. F# uo vo pMeshG F F# eval : nu F rows 1 - : nv F cols 1 - : 13 mat np uo length : r.np 1 np range : r.nv 1 nv 1 - (range : 13 mat M 03 matrix eval : c 13 range : 12 mat u 1 nu range v 1 nv range r M rows : k r r.np + : 12 mat M k c el F uo k r - el u + vo k r - el v + el c el : 21 line for r M rows : j r r.nv + : 12 mat M j c el M u 1 - (np nv * nv + 1 - (* np nv 1 - (* + j r - 1 - (np * - c el : 31 line for M 51 line :

Short hands

F# U V pTMesh X Y 12 mat U V pGrid : F# X Y pGrid F# X Y pTGrid 0010115 mat 0110015 mat pTMeshG 21 line :

F# U V pMesh F# U V pGrid 0011015 mat 0110015 mat pMeshG 11 line :

F# U V pWXMesh X Y 12 mat U V pGrid : X Y 12 mat X 1 el X stack Y 1 el Y stack 12 mat eval : F# X Y pGrid 0012 mat 1112 mat pMeshG 31 line :

F# U V pWYMesh X Y 12 mat U V pGrid : X Y 12 mat X 1 el X stack Y 1 el Y stack 12 mat eval : F# X Y pGrid transpose 0012 mat 1112 mat pMeshG 31 line :

<content> pContour </content>

¦AuxIPE 0931061551212515610390116 mat :

¦AuxIPΔ 0110 0.5 1100 0.5 25 mat :

¦AuxIPBin 124841 mat :

¦AuxIPT 0112211331122111042434344323424000000200003000000000030000400000416 mat :

¦AuxIPB 0110011001100110001111000011110000001111111100000101010110101010416 mat :

Contour Plot FS# U V NL pContour X Y 12 mat U V pGrid : Z FS# X Y pRGrid : 12 mat h 14 range : CL 01 matrix eval : 12 mat NL IsMatrix L NL eval : L Z max Z min NL 2 + pR 2 NL 1 + (range el eval : if q 1 L length range ZL Z L q el - eval : M 0 : m 0 : v 0 : P 0 : z 0 : 16 mat r 1 X length 2 - (3 range c 1 Y length 2 - (3 range Δx 022014 mat r + : Δy 220014 mat c + : z h el ZL Δx h el Δy h el el : 13 mat n 1 z 0 < vectorize ¦AuxIPBin * + eval : ¦AuxIPE n el 0 ≡ continue ZL r 1 + c 1 + el 0 < (n 11 ≡ (n 6 ≡ (| (& n n 11 ≡ (6 * n 6 ≡ (11 * + eval : 0 if x X Δx el : y Y Δy el : 12 mat k 1312 mat ¦AuxIPT k n el 0 ≠ s ¦AuxIPT k n el ¦AuxIPT k 1 + n el 12 mat : v s el ¦AuxIPB s n el 0 ≠ j 1 s 4 mod + : x s el y s el 12 mat x j el x s el - y j el y s el - 12 mat z j el z s el - / z s el * - eval 21 line 0 if : M m m 1 + : el v s 1 el el v s 2 el el stack eval : 31 line continue if for 31 line if 31 line for for CL CL M stack eval : 31 line for CL 51 line :

Contour Plot FS# U V NL pContour X Y 12 mat U V pGrid eval : Z FS# X Y pRGrid eval : 12 mat h 14 range : CL 0 : 12 mat NL IsMatrix L NL eval : L Z max Z min NL 2 + pR 2 NL 1 + (range el eval : if q 1 L length range ZL Z L q el - eval : M 0 : m 0 : v 0 : P 0 : z 0 : 16 mat r 1 X length 2 - (3 range c 1 Y length 2 - (3 range Δx 022014 mat r + : Δy 220014 mat c + : z h el ZL Δx h el Δy h el el : 13 mat n 1 z 0 < vectorize ¦AuxIPBin * + eval : ¦AuxIPE n el 0 ≡ continue ZL r 1 + c 1 + el 0 < (n 11 ≡ (n 6 ≡ (| (& n n 11 ≡ (6 * n 6 ≡ (11 * + eval : 0 if x X Δx el : y Y Δy el : 12 mat k 1312 mat ¦AuxIPT k n el 0 ≠ s ¦AuxIPT k n el ¦AuxIPT k 1 + n el 12 mat : v s el ¦AuxIPB s n el 0 ≠ j 1 s 4 mod + : x s el y s el 12 mat x j el x s el - y j el y s el - 12 mat z j el z s el - / z s el * - eval 21 line 0 if : M m m 1 + : el v s 1 el el v s 2 el el stack eval : iter 1 pCycleC range M m m 1 + : el 0012 mat : for 41 line continue if for 31 line if 31 line for for CL q el M mat2sys.1 : iter 1 pCycleC range cCL pCycleC 1 + (q * pCycleC - iter + el 0012 mat : for 41 line for CL q 1 + el 0012 mat : mat2sys.1 51 line :

Z X Y pContour Z X Y 011 mat pContour 11 line :

<content> pZ & FillContour </content>

FS# U V g pFillContour X Y 12 mat U V pGrid : M FS# X Y pRGrid : 12 mat X Y M g pClr pFill2 21 line :

FS# U V g pZ X Y 12 mat U V pGrid : Z FS# X Y pZGrid : 12 mat X Y Z Arg vectorize g pClr pFill2 21 line :

X Y a pFill2 nx X length : ny Y length : 12 mat r 1 nx 1 - (range : c 1 ny 1 - (range : S 0 : 13 mat S r c el X r el Y c el X r el Y c 1 + el X r 1 + el Y c 1 + el X r 1 + el Y c el 42 mat a r 1 - (ny * c + el pPolygon : S 1 S length range el 11 mat 41 line :

FS# U V pZ X Y 12 mat U V pGrid : nx X length : ny Y length : Sc 0 : Sr 0 : 14 mat r 1 nx range : c 1 ny range : 12 mat Zc FS# X 1 el X nx el pZ_refine nx * 1 - pR Y pZGrid : Sc c el Zc c col ReIm pZ_blue pLine eval : Zr FS# X Y 1 el Y ny el pZ_refine ny * 1 - pR pZGrid : Sr r el Zr r row transpose ReIm pZ_red pLine eval : Sc Sr stack 11 mat eval 81 line :

pZ_blue #7F7F0000 :

pZ_red #7F007FFF :

pZ_refine 8 :

pZ polyline :

spline

polyline

bezier

R γ pD3 R cols 3 > X Y Z 13 mat R 2 col R 3 col R 4 col 13 mat eval : X Y Z 13 mat R 1 col R 2 col R 3 col 13 mat eval : if γ2 γ 13 range 12 range el eval : M X Y Z augment eval : M2 M γ2 * eval : 13 mat mX2 mY2 12 mat M2 1 col min abs M2 2 col min abs 12 mat eval : B M pD3% pBox eval : m a b b2 m2 m γ2 * eval : m2 1 col a 1 pD3% + (* - m2 2 col b 1 pD3% + (* - augment eval 21 line : b B 11 el B 21 el B 31 el B 12 el B 21 el B 31 el B 12 el B 22 el B 31 el B 11 el B 22 el B 31 el 43 mat B 11 el B 21 el B 31 el B 12 el B 21 el B 31 el B 12 el B 21 el B 32 el B 11 el B 21 el B 32 el 43 mat B 11 el B 21 el B 31 el B 11 el B 21 el B 32 el B 11 el B 22 el B 32 el B 11 el B 22 el B 31 el 43 mat 13 mat : M2 X Y B 31 el augment 0 mY2 b2 X B 21 el Z augment mX2 0 b2 B 11 el Y Z augment mX2 - 0 b2 b 1 el 0 mY2 b2 pD3Clr pPolygon 11 mat b 2 el mX2 0 b2 pD3Clr pPolygon 11 mat b 3 el mX2 - 0 b2 pD3Clr pPolygon 11 mat 31 mat M γ pAxis 11 mat M2 1 row . 6 yellow augment 71 sys 71 line :

pD3Clr #4466FF44 :

pD3% 5 % * :

<content> pStem </content>

X Y c pStem X IsMatrix S X 1 el 0 X 1 el Y 1 el 22 mat : k 2 : k X length ≤ k k 1 + : S S X k 1 - el 012 mat X k el 012 mat X k el Y k el 12 mat stack eval : for S X Y c augment 17 mat mat2sys.1 31 line Y X Y pRGrid c pStem if 11 line :

X Y pStem X Y o pStem 11 line :

FS a# X# Y# ν μ εx εy δx MI fit.LM X X# eval : Y Y# eval : 12 mat X cols 1 ≡ co 1 : co 1 X cols range : if m X rows : n a# length : r 1 m range : c 1 n range : 14 mat a a# c el : Id n identity eval : 12 mat Na ∞ : upd 1 : FX 0 : Fa 0 : Fb 0 : J 0 : 16 mat iter 1 MI range upd 1 ≡ FX r el FS X r co el a pFeval : J r c el FS X r co el a δx Id c col (* + pFeval FX r el - δx / eval : Fa r el FX r el Y r el - eval : H J transpose J * eval : 12 mat 31 line continue if s H μ Id * + (1 -^J transpose Fa * (* eval : b a s - : Fb r el FS X r co el a pFeval Y r el - eval : Nb Fb norme : 13 mat μ sqrt Nb * Na < s normi εx a normi * < break a b : Fa Fb : Na Nb : μ μ ν / : upd 1 : 15 mat if μ ν μ * : upd 0 : 12 mat if 41 line for a eval 71 line :

FS# p X Y fit.LM FS# p X Y ν.LM μ.LM εx εy δx MaxIters fit.LM 11 line :

x y franke 0.75 e 9 x * 2 - (2^9 y * 2 - (2^+ 4 / -^* 0.75 e 9 x * 1 + (2^49 / - 9 y * 1 + 10 / -^* + 0.5 e 9 x * 7 - (2^9 y * 3 - (2^+ 4 / -^* + 0.2 e 9 x * 4 - (2^- 9 y * 7 - (2^-^* - :

εx 105 -^:

εy 107 -^:

MaxIters 100 :

ν.LM 10 :

μ.LM 0.01 ν.LM * :

δx 100 εx * :

<content> PolyFit </content>

x y n PolyFit m x length : r 1 m range : c 1 n 1 + (range : A m n 1 + matrix 1 + : 14 mat A r c el x r el x r el 0 ≡ (c 1 ≡ (& (+ (c 1 -^: A transpose A * (1 -^A transpose y * (* eval 31 line :

You can try to substitute the last line with someone of the following, from dislin plugin, which are more robust and general

A y dn_LinAlgLLS_COF

In matlab: (A\y).'

A y dn_LinAlgLLS_SVD

A y dn_LinAlgLLS_QRorLQ

A is the Vandermonde matrix. The code

x r el 0 ≡ (c 1 ≡ (&

is for avoid the SMath error 0^0, Uncertainty.

p x PolyVal P 0 : k 0 : 12 mat c p P P c x k^vectorize * + eval : k k 1 + : 12 mat for P 31 line :

Horner Method

p k el x k 1 -^vectorize * k 1 n sum

X Y R.Pearson Ξ Ψ 12 mat X Y 12 mat 1 X length / X sum Y sum 12 mat * - : Ξ Ψ * vectorize sum Ξ 2^vectorize sum (Ψ 2^vectorize sum (* sqrt / 21 line :

F X Y n pPolyFit C X Y n PolyFit : r 1101 range : S X 1 el Y 1 el 12 mat : 13 mat YR 0 : XR X min X max 100 pR : 12 mat k 2 X length range S S X k 1 - el 012 mat X k el 012 mat X k el Y k el 12 mat stack eval : for 11 line XR YR r el C XR r el PolyVal : augment S X Y o augment X C X PolyVal x augment 41 sys 41 line :

M n pPolyFit M M 1 col M 2 col n pPolyFit 11 line :

X Y n R2 n 0 ≡ Not defined Y X Y n PolyFit X PolyVal R.Pearson (2^if 11 line :

pMesh.np 5 :

NaN 1 :

∞ 1010^:

pCycleC 5 :

pShowAxis 1 :

γ 120 ° * 30 ° * pView :

CMap Jet 100 0.8 pCMap :

Curve Fitting

<content> Stress-Strain Fit </content> <content> Data: XY </content>

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94.4577846600000 2.7648000720000 94.4625970100000 2.7695956230000 94.4745964700000 2.7743957040000 94.4778884700000 2.7791955470000 94.4796601100000 2.7839956280000 94.4796475500000 2.7864000800000 94.4827950500000 2.7911999230000 94.4823615600000 2.7960000040000 94.4833227800000 2.8007998470000 94.4881100000000 2.8055956360000 94.4875759900000 2.8103957180000 94.4878587000000 2.8127999310000 94.4799114100000 2.8176000120000 94.4763241400000 2.8223998550000 94.4740184800000 2.8271956440000 94.4733839600000 2.8296000960000 94.4714426800000 2.8343999390000 94.4667497000000 2.8392000200000 94.4622263400000 2.8439998630000 94.4579354400000 2.8487999440000 94.4529094900000 2.8535954950000 94.4406398800000 2.8583998680000 94.4347218200000 2.8631999490000 94.4252416200000 2.8680000310000 94.4139395100000 2.8727955820000 94.3960220000000 2.8775956630000 94.3861397200000 2.8824000360000 94.3594707600000 2.8871998790000 94.3347556500000 2.8919999600000 94.3117493500000 2.8968000410000 94.2844835700000 2.9015998840000 94.2604909300000 2.9063999650000 94.2232109300000 2.9112000470000 94.1838577300000 2.9159998890000 94.1301805700000 2.9231956010000 94.0653960500000 2.9279999730000 93.9869912100000 2.9351956840000 93.8739073000000 2.9375998970000 93.7519777100000 2.9447956090000 93.6202977800000 2.9472000600000 93.5020874000000 2.9472000600000 93.3814645600000 2.9495956900000 92.7670922400000 11842 mat :

Stress - Strain Data

X Y 12 mat XY MCols :

XY rows 1184

k 1 XY rows range :

Target: interpolate at

εo 1.7 :

σ.max Y max eval :

PolyFit

n 2 :

C X Y n PolyFit : 1.0996 76.2994 15.6412 - 31 mat

c = Intercept
b = Coeff of x
a = Coeff of x^2

σ a ε 2^* b ε * + c + ≡

plot XY X C X PolyVal augment εo 11 mat C εo PolyVal 11 mat pStem 31 sys :

plot

C εo PolyVal 85.6056

X Y n R2 0.9984

GenFit

Mathematical model and guess value

ε β σ β 1 el β 2 el e β 3 el ε * -^* - :

β.guess σ.max σ.max 131 mat :

SMath solution

βo σ β.guess X Y fit.LM : 104.6755 107.6579 0.9414 31 mat

Final stress
Hardening parameter
Exponent

plot XY X σ k el X k el βo σ : augment εo 11 mat εo βo σ 11 mat pStem 31 sys :

plot

εo βo σ 82.9473

GenFit II

Mathematical model and guess value

ε β σ.2 β 1 el β 2 el ε abs β 3 el^* + :

β.guess 0 σ.max 131 mat :

SMath solution

βo σ.2 β.guess X Y fit.LM : 15.3346 - 74.4498 0.4569 31 mat

Yield stress
Hardening parameter
Exponent

plot XY X σ k el X k el βo σ.2 : augment εo 11 mat εo βo σ.2 11 mat pStem 31 sys :

plot

εo βo σ.2 79.539

Other models

Can try with Hollomon or Ludwigson models

<content> PolyFit </content>

Empirical Data

Data 0.5 3.0 1.1 1.2 1.5 2.3 2.1 0.2 2.3 1.8 52 mat :

X Y 12 mat Data MCols :

Target: interpolate at

xo 1.7 :

Horinzontal Line Fit

n 0 :

Data n pPolyFit

C X Y n PolyFit : 1.7

y constant ≡

C xo PolyVal 1.7

Y sum Y length / 1.7

X Y n R2 Not defined

Lineal Fit

n 1 :

Data n pPolyFit

C X Y n PolyFit : 3.0333 0.8889 - 21 mat

b = Intercept
a = Slope

C xo PolyVal 1.5222

y a x * b + ≡

X Y n R2 0.3743

Quadratic Polyfit

n 2 :

Data n pPolyFit

C X Y n PolyFit : 4.2513 3.0751 - 0.7686 31 mat

c = Intercept
b = Coeff of x
a = Coeff of x^2

C xo PolyVal 1.2448

y a x 2^* b x * + c + ≡

X Y n R2 0.4493

Matlab and Excel PolyFit example

This example can help identifiying the same values in SMath, Excel and Matlab

x f x sin :

X 09 range 9 / 4 π * * :

Y X f vectorize :

Ten points over a sinusoid

n 6 :

PolyFit degree

C X Y n PolyFit :

X Y n R2 0.9057

C 0.034 - 2.8327 1.9611 - 0.4452 0.0407 - 0.0013 3.47 1013 -^* - 71 mat

Matlab screenshoot

Excel screenshot

SMath plot

X Y augment n pPolyFit <content> GenFit </content>

GenFit

Example from Mathcad help:

http://support.ptc.com/help/mathcad/en/index.html#page/PTC_Mathcad_Help%2Fexample_nonlinear_regression1.html%23wwID0ERQTR

Empirical data

Y 2.513 2.044 1.668 1.366 1.123 0.927 0.768 0.639 0.534 0.448 0.378 0.32 0.272 0.232 0.2 0.172 0.149 0.13 0.114 0.1 0.088 0.078 0.07 0.062 stack :

X 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 stack :

Mathematical model and guess value

x β f β 1 el e β 2 el x * -^* β 3 el e β 4 el x * -^* + β 5 el e β 6 el x * -^* + :

β.guess 0.5 0.7 3.6 4.2 4.0 6.3 61 mat :

SMath solution

β.SMath f β.guess X Y fit.LM :

NonLinear solver plugin solution

β target r 1 X length range : Δ r el X r el β f Y r el - (: norme 21 line :

β.NM β target β.guess NelderMead :

Maxima plugin solution

β.Maxi β.Maxi β1 β2 β3 β4 β5 β6 16 mat transpose : X Y augment x y 21 sys y x β.Maxi f ≡ β.Maxi β.guess Fit Assign 21 line :

Comparing solutions

β.SMath 0.1018 1.0229 0.9849 3.1335 1.4264 5.1048 61 mat

β.NM 1.0882 - 3.2112 2.8646 3.5727 0.6783 3.6097 61 mat

β.Maxi 0.3598 1.4793 1.1584 3.9027 1.0525 6.3523 61 sys

Mathcad solution

plot x β.SMath f x β.NM f x β.Maxi f X Y o augment 41 sys :

plot <content> Multivariate genfit </content>

Multivariate GenFit

Generate some bivariate data with noise

xo yo f x xo 0.1 noise + : y yo 0.1 noise + : 12 mat 7 x 2^* 6 y 2^* - 5 x * + 0.5 noise + 21 line :

B 1 - 11 - 122 mat :

N 202021 mat :

G f B N pGrid :

Theoretical model

u β F β 1 el u 1 el 2^* β 2 el u 2 el 2^* + β 3 el u 1 el * + :

β.guess 111 stack :

Arrange the data

r 1 G length range :

c 13 range :

XYZ r c el G r el c el :

Fit the model to the data

βo F β.guess XYZ r 1212 mat el XYZ r 3 el fit.LM : 7.0289 5.8474 - 5.16 31 mat

CMap Jet 100 0.8 pCMap : γ 30 ° * - 45 ° * pView : x y Fo x y stack βo F : plot Fo B N pMesh CMap γ pShow XYZ γ 13 range 1212 mat el * o 3 augment 21 sys : 41 line

plot

Alvaro