Advanced Search

dharr

Dr. David Harrington

8205 Reputation

22 Badges

20 years, 337 days
University of Victoria
Professor or university staff
Victoria, British Columbia, Canada

Social Networks and Content at Maplesoft.com

Maple Application Center
Contact Dr. David Harrington
I am a retired professor of chemistry at the University of Victoria, BC, Canada. My research areas are electrochemistry and surface science. I have been a user of Maple since about 1990.

MaplePrimes Activity


These are answers submitted by dharr

size...

dharr 8205
March 19 2024
1 1

The default appears to be 500x500 pixels. If you add the plot option size=[1000,1000] then the exported file will be 1000x1000 pixels.

Table of Contents...

dharr 8205
March 17 2024
4 3

You need to set up a Table of Contents to load into the help database with HelpTools:-TableOfContents:-Load(TOC,helpdbname);

In the TOC, you set a priority for each entry, so make your overview the highest priority

See here on Mapleprimes for an example.

implicitplot3d...

dharr 8205
March 15 2024
1 2

Since you only know M via the equation and not explicitly, implicitplot3d is appropriate.

implicitplot3d.mw

one way...

dharr 8205
March 14 2024
1 1

Here's a way that only generates valid possibilities for i, j, k. I'm assuming each of i, j, k is in the range 1..n.

> 

restart
> 

n := 5

5

Compositions of k with 3 parts (sum to k). Need ones that add to k=3, 4, .. n

> 

comps := table([seq(k = Iterator:-Composition(k, parts = 3), k = 3 .. n)])

table( [( 3 ) = _m2362689082336, ( 4 ) = _m2362689084160, ( 5 ) = _m2362722715232 ] )

Check possibilities

> 

for k from 3 to n do lprint(cat("Add to ", k, ":")); Print(comps[k]) end do

Add to 3:
1 1 1
Add to 4:
2 1 1
1 1 2
1 2 1
Add to 5:
3 1 1
2 1 2
2 2 1
1 1 3
1 2 2
1 3 1

Calculate the required sum

> 

add(add(1/mul(comp), `in`(comp, comps[k])), k = 3 .. n)

17/4

So work this out for the first few n.
If we are going to do all of them we can avoid duplicate iterators and incrementally accumulate sums.

> 

s := 0; for n to 10 do comps := Iterator:-Composition(n, parts = 3); s := s+add(1/mul(comp), `in`(comp, comps)); print(n, s) end do

1, 0

2, 0

3, 1

4, 5/2

5, 17/4

6, 49/8

7, 967/120

8, 801/80

9, 4523/378

10, 84095/6048

> 

NULL

Download compositions.mw

First one to StandardFunctions...

dharr 8205
March 13 2024
2 3
> 

restart;
> 

q:=hypergeom([1, 1, 3/2], [5/2, 5/2, 3, 3], -(x/2)**2);

hypergeom([1, 1, 3/2], [5/2, 5/2, 3, 3], -(1/4)*x^2)

Note sure about the assumptions - choose one root.
Often conversions work when the argument is simple.

> 

itr:=u=-x^2/4;
tr:=x=solve(u=-x^2/4,x)[1];

u = -(1/4)*x^2

x = 2*(-u)^(1/2)

> 

qu:=PDEtools:-dchange(tr,q);

hypergeom([1, 1, 3/2], [5/2, 5/2, 3, 3], u)

> 

convert(qu, StandardFunctions);
q2:=PDEtools:-dchange(itr,%,simplify);

-(-108*u+27)/u^2+(1/2)*(-18*u^(1/2)+27)*exp(2*u^(1/2))/u^2+(1/2)*(18*u^(1/2)+27)*exp(-2*u^(1/2))/u^2+(18*u-36*u^(1/2)+9)*(-gamma-ln(-2*u^(1/2))-Ei(1, -2*u^(1/2)))/u^2+(18*u+36*u^(1/2)+9)*(-gamma-ln(2*u^(1/2))-Ei(1, 2*u^(1/2)))/u^2

36*(-12+2*(x^2+4*(-x^2)^(1/2)-2)*Ei(1, -(-x^2)^(1/2))+2*(x^2+4*(-x^2)^(1/2)-2)*ln(-(-x^2)^(1/2))+2*(3+(-x^2)^(1/2))*exp(-(-x^2)^(1/2))+2*(x^2-4*(-x^2)^(1/2)-2)*Ei(1, (-x^2)^(1/2))+2*(3-(-x^2)^(1/2))*exp((-x^2)^(1/2))-4*ln(-x^2)*(-x^2)^(1/2)+(x^2-2)*ln(-x^2)+4*(-3+gamma)*x^2-8*gamma)/x^4

> 

NULL

Download dchange.mw

Dirac...

dharr 8205
March 13 2024
2 2

If I understand what you want, Dirac can do this for you. Note that aside frim the infinitesimal time interval aspect @C_R mentioned, the ifelse immediately finds that t is not 1 and so Inf =0 after the ifelse statement is executed.

> 

restart;
> 

k12:=0.0452821;
k21:=0.0641682;
k1x:=0.00426118;
Dose:=484;
#Inj:=ifelse(t=1,Dose,0);

ode_sys:=diff(SA2(t),t)=SA1(t)*k12-SA2(t)*k21,
         diff(SA1(t),t)=Dose*Dirac(t-1)-k1x*SA1(t);

ics:=SA2(0)=0,SA1(0)=0;

0.452821e-1

0.641682e-1

0.426118e-2

484

diff(SA2(t), t) = 0.452821e-1*SA1(t)-0.641682e-1*SA2(t), diff(SA1(t), t) = 484*Dirac(t-1)-0.426118e-2*SA1(t)

SA2(0) = 0, SA1(0) = 0

> 

Solution1:=dsolve({ode_sys,ics},{SA1(t),SA2(t)},type=numeric);

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 28, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([Array(1..5, 1..21, {(1, 1) = 1.0, (1, 2) = 2.0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .999999800002, (1, 6) = .0, (1, 7) = 1.0, (1, 8) = undefined, (1, 9) = undefined, (1, 10) = 1.0, (1, 11) = undefined, (1, 12) = undefined, (1, 13) = undefined, (1, 14) = undefined, (1, 15) = undefined, (1, 16) = undefined, (1, 17) = undefined, (1, 18) = undefined, (1, 19) = undefined, (1, 20) = undefined, (1, 21) = undefined, (2, 1) = 2.0, (2, 2) = 2.0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .999999999998, (2, 6) = .0, (2, 7) = 1.0, (2, 8) = undefined, (2, 9) = undefined, (2, 10) = 1.0, (2, 11) = undefined, (2, 12) = undefined, (2, 13) = undefined, (2, 14) = undefined, (2, 15) = undefined, (2, 16) = undefined, (2, 17) = undefined, (2, 18) = undefined, (2, 19) = undefined, (2, 20) = undefined, (2, 21) = undefined, (3, 1) = 3.0, (3, 2) = 2.0, (3, 3) = .0, (3, 4) = .0, (3, 5) = 1.000000000002, (3, 6) = .0, (3, 7) = 1.0, (3, 8) = undefined, (3, 9) = undefined, (3, 10) = 1.0, (3, 11) = undefined, (3, 12) = undefined, (3, 13) = undefined, (3, 14) = undefined, (3, 15) = undefined, (3, 16) = undefined, (3, 17) = undefined, (3, 18) = undefined, (3, 19) = undefined, (3, 20) = undefined, (3, 21) = undefined, (4, 1) = 4.0, (4, 2) = 2.0, (4, 3) = .0, (4, 4) = .0, (4, 5) = 1.000000199998, (4, 6) = .0, (4, 7) = 1.0, (4, 8) = undefined, (4, 9) = undefined, (4, 10) = 1.0, (4, 11) = undefined, (4, 12) = undefined, (4, 13) = undefined, (4, 14) = undefined, (4, 15) = undefined, (4, 16) = undefined, (4, 17) = undefined, (4, 18) = undefined, (4, 19) = undefined, (4, 20) = undefined, (4, 21) = undefined, (5, 1) = 4.0, (5, 2) = .0, (5, 3) = 100.0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (5, 8) = undefined, (5, 9) = undefined, (5, 10) = 0.10e-6, (5, 11) = undefined, (5, 12) = .0, (5, 13) = undefined, (5, 14) = .0, (5, 15) = .0, (5, 16) = undefined, (5, 17) = undefined, (5, 18) = undefined, (5, 19) = undefined, (5, 20) = undefined, (5, 21) = undefined}, datatype = float[8], order = C_order), proc (t, Y, Ypre, n, EA) EA[1, 8+2*n] := 1; EA[2, 8+2*n] := 1; EA[3, 8+2*n] := 1; EA[4, 8+2*n] := 1; 0 end proc, proc (e, t, Y, Ypre) return 0 end proc, Array(1..4, 1..2, {(1, 1) = undefined, (1, 2) = undefined, (2, 1) = undefined, (2, 2) = undefined, (3, 1) = undefined, (3, 2) = undefined, (4, 1) = undefined, (4, 2) = undefined}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..65, {(1) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 4, (17) = 0, (18) = 29, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0, (64) = -1, (65) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.10e-5, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..2, {(1) = .1, (2) = .1}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = .0, (1, 2) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = SA1(t), Y[2] = SA2(t)]`; YP[1] := 484*piecewise(X < 4999999/5000000, 0, X < 1, -24999995000000+25000000000000*X, X < 5000001/5000000, 25000005000000-25000000000000*X, 0)-0.426118e-2*Y[1]; YP[2] := 0.452821e-1*Y[1]-0.641682e-1*Y[2]; 0 end proc, -1, 0, 0, 0, 0, proc (t, Y, Ypre, n, EA) EA[1, 8+2*n] := 1; EA[2, 8+2*n] := 1; EA[3, 8+2*n] := 1; EA[4, 8+2*n] := 1; 0 end proc, proc (e, t, Y, Ypre) return 0 end proc, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = SA1(t), Y[2] = SA2(t)]`; YP[1] := 484*piecewise(X < 4999999/5000000, 0, X < 1, -24999995000000+25000000000000*X, X < 5000001/5000000, 25000005000000-25000000000000*X, 0)-0.426118e-2*Y[1]; YP[2] := 0.452821e-1*Y[1]-0.641682e-1*Y[2]; 0 end proc, -1, 0, 0, 0, 0, proc (t, Y, Ypre, n, EA) EA[1, 8+2*n] := 1; EA[2, 8+2*n] := 1; EA[3, 8+2*n] := 1; EA[4, 8+2*n] := 1; 0 end proc, proc (e, t, Y, Ypre) return 0 end proc, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 27 ) = (""), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0), ( 28 ) = (0) ] )) ] ); _y0 := Array(0..2, {(1) = 0., (2) = 0.}); _vmap := array( 1 .. 2, [( 1 ) = (1), ( 2 ) = (2) ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch: end try else try procname(procname("right")) catch: end try end if end if end if; return elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch: end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch: end try end if end if end if; return elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if elif type(_xin, `=`) and lhs(_xin) = "setdatacallback" then if not type(rhs(_xin), 'nonegint') then error "data callback must be a nonnegative integer (address)" end if; _dtbl[1][28] := rhs(_xin) else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [t, SA1(t), SA2(t)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error end try end proc

> 

plots:-odeplot(Solution1,[[t,SA1(t),color="red",thickness=1],[t,SA2(t),color="blue",thickness=1]],t=0..50,labels=["Time t","Salicylic acid"],labeldirections=[horizontal,vertical]);

> 

NULL

Download Dirac.mw

Biorthogonality...

dharr 8205
March 13 2024
2 2

@mehdi jafari You asked about the relationship between left and right eigenvectors, and getting a diagonal matrix from them. They are biorthogonal, meaning that left eigenvectors are orthogonal to right eigenvectors corresponding to different eigenvalues. So the product of the matrices of left eigenvectors (as row vectors) and right eigenvectors (as column vectors) is diagonal.

[Edit - 2023 version uploaded] Rerun the worksheet in your version of Maple; the rtable output from Maple 2024 doesn't display correctly in earlier versions (or here on Mapleprimes, SCR submitted).

> 

restart
> 

with(LinearAlgebra)
> 

A := Matrix(3, 3, [1, 0, 1, 1, 1, 0, 2, 0, 1])

Matrix(%id = 36893490995083995364)

Right (column) eigenvectors corresponding to increasing eigenvalues (any order for the eigenvalues will do, so long as it is consistent for left and right eigenvectors)

> 

revs, rvecs1 := Eigenvectors(A); evs, rperm := sort(revs, output = [sorted, permutation], key = evalf); rvecs := rvecs1[() .. (), rperm]

evs, rperm := Vector(3, {(1) = 1-sqrt(2), (2) = 1, (3) = 1+sqrt(2)}), [3, 1, 2]

Matrix(%id = 36893490995083958140)

Left (row) eigenvectors corresponding to increasing eigenvals

> 

levs, lvecs1 := Eigenvectors(LinearAlgebra:-Transpose(A)); evs, lperm := sort(levs, output = [sorted, permutation], key = evalf); lvecs := lvecs1*LinearAlgebra:-Transpose([() .. (), lperm])

evs, lperm := Vector(3, {(1) = 1-sqrt(2), (2) = 1, (3) = 1+sqrt(2)}), [2, 3, 1]

Matrix(%id = 36893490995083988852)

Biorthogonality means that left eigenvectors are orthogonal to right eigenvectors corresponding to different eigenvalues.

e.g, left eigenvector for 1 is orthogonal to right eigenvalues for 1-srqt(2) and 1+sqrt(2)

> 

lvecs[1, () .. ()], rvecs[() .. (), 2]; lvecs[1, () .. ()].rvecs[() .. (), 2]; lvecs[1, () .. ()].rvecs[() .. (), 3]

Vector[row](%id = 36893490995083967172), Vector[column](%id = 36893490995083967292)

0

0

All possibilities

> 

lvecs.rvecs

Matrix(%id = 36893490995083956340)

> 

NULL

Download Biorthogonal.mw

 

normalization...

dharr 8205
March 12 2024
3 0

The eigenvectors from Eigenvectors are arbitrarily scaled. In this case they happen to be normalized but differ in signs. Once you account for this (and a forgotten square root), there is agreement. Notice you are also relying on the orders of the eigenvalues and singular values being consistent.

 

WHY DOES THE TRANSPOSE OF Vt DETERMINED BY THE FUNCTION SingularValues NOT AGREE WITH THE evectors CALCULATED BY THE FUNCTION Eigenvectors?

 

> 

restart:
with(LinearAlgebra):
> 

X := Matrix([[8.79,9.93,9.83,5.45,3.16],
           [6.11,6.91,5.04,-0.27,7.98],
           [-9.15,-7.93,4.86,4.85,3.01],
           [9.57,1.64,8.83,0.74,5.80],
           [-3.49,4.02,9.80,10.00,4.27],
           [9.84,0.15,-8.99,-6.02,-5.31]],
           datatype=float[8],order=Fortran_order);

Matrix(6, 5, {(1, 1) = 8.79, (1, 2) = 9.93, (1, 3) = 9.83, (1, 4) = 5.45, (1, 5) = 3.16, (2, 1) = 6.11, (2, 2) = 6.91, (2, 3) = 5.04, (2, 4) = -.27, (2, 5) = 7.98, (3, 1) = -9.15, (3, 2) = -7.93, (3, 3) = 4.86, (3, 4) = 4.85, (3, 5) = 3.01, (4, 1) = 9.57, (4, 2) = 1.64, (4, 3) = 8.83, (4, 4) = .74, (4, 5) = 5.8, (5, 1) = -3.49, (5, 2) = 4.02, (5, 3) = 9.8, (5, 4) = 10.0, (5, 5) = 4.27, (6, 1) = 9.84, (6, 2) = .15, (6, 3) = -8.99, (6, 4) = -6.02, (6, 5) = -5.31})

 (1)
> 

U,S,Vt:= SingularValues(X, output = ['U', 'S', 'Vt'],thin=true)

U, S, Vt := Matrix(6, 5, {(1, 1) = -.591142376412437, (1, 2) = .263167814714055, (1, 3) = .355430173862827, (1, 4) = .314264362726927, (1, 5) = .229938315364748, (2, 1) = -.397566794202426, (2, 2) = .243799027926330, (2, 3) = -.222390000685446, (2, 4) = -.753466150953458, (2, 5) = -.363589686697497, (3, 1) = -0.334789690624459e-1, (3, 2) = -.600272580693583, (3, 3) = -.450839268922308, (3, 4) = .233449657244714, (3, 5) = -.305475732747932, (4, 1) = -.429706903137018, (4, 2) = .236166806281125, (4, 3) = -.685862863873811, (4, 4) = .331860018200310, (4, 5) = .164927634884511, (5, 1) = -.469747921566658, (5, 2) = -.350891398883703, (5, 3) = .387444603099673, (5, 4) = .158735559582156, (5, 5) = -.518257437353535, (6, 1) = .293358758464403, (6, 2) = .576262119133891, (6, 3) = -0.208529179808710e-1, (6, 4) = .379077667060160, (6, 5) = -.652551600592398}), Vector(6, {(1) = 27.4687324182218, (2) = 22.6431850097747, (3) = 8.55838822848258, (4) = 5.98572320151213, (5) = 2.01489965871576, (6) = 0.}), Matrix(5, 5, {(1, 1) = -.251382792720498, (1, 2) = -.396845551776930, (1, 3) = -.692151007470363, (1, 4) = -.366170444772230, (1, 5) = -.407635238653352, (2, 1) = .814836686086339, (2, 2) = .358661500188002, (2, 3) = -.248888011159285, (2, 4) = -.368593537944618, (2, 5) = -0.979625692668875e-1, (3, 1) = -.260618505584221, (3, 2) = .700768209407253, (3, 3) = -.220811446720437, (3, 4) = .385938483188542, (3, 5) = -.493250142851024, (4, 1) = .396723777130597, (4, 2) = -.450711241216643, (4, 3) = .251321149693754, (4, 4) = .434248601436671, (4, 5) = -.622684072035804, (5, 1) = -.218027763686546, (5, 2) = .140209949871121, (5, 3) = .589119449239943, (5, 4) = -.626528250364817, (5, 5) = -.439551692342332})

 (2)
> 

SDM:= DiagonalMatrix(S[1..5],5,5)

Matrix(5, 5, {(1, 1) = 27.46873241822184, (1, 2) = 0., (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (2, 1) = 0., (2, 2) = 22.643185009774694, (2, 3) = 0., (2, 4) = 0., (2, 5) = 0., (3, 1) = 0., (3, 2) = 0., (3, 3) = 8.558388228482578, (3, 4) = 0., (3, 5) = 0., (4, 1) = 0., (4, 2) = 0., (4, 3) = 0., (4, 4) = 5.985723201512132, (4, 5) = 0., (5, 1) = 0., (5, 2) = 0., (5, 3) = 0., (5, 4) = 0., (5, 5) = 2.014899658715757})

 (3)

THIS EQUALS TO ORIGINAL X MATRIX

 

> 

U.SDM.Vt

Matrix(6, 5, {(1, 1) = 8.789999999999997, (1, 2) = 9.93, (1, 3) = 9.829999999999995, (1, 4) = 5.449999999999993, (1, 5) = 3.159999999999998, (2, 1) = 6.1099999999999985, (2, 2) = 6.9099999999999975, (2, 3) = 5.0399999999999965, (2, 4) = -.26999999999999996, (2, 5) = 7.980000000000001, (3, 1) = -9.14999999999999, (3, 2) = -7.930000000000001, (3, 3) = 4.859999999999987, (3, 4) = 4.849999999999992, (3, 5) = 3.009999999999995, (4, 1) = 9.569999999999997, (4, 2) = 1.6399999999999977, (4, 3) = 8.82999999999999, (4, 4) = .7399999999999956, (4, 5) = 5.799999999999994, (5, 1) = -3.489999999999992, (5, 2) = 4.019999999999998, (5, 3) = 9.799999999999985, (5, 4) = 9.999999999999988, (5, 5) = 4.269999999999993, (6, 1) = 9.83999999999999, (6, 2) = .15000000000000033, (6, 3) = -8.989999999999982, (6, 4) = -6.0199999999999925, (6, 5) = -5.309999999999993})

 (4)
> 

X -~ U.SDM.Vt

Matrix(6, 5, {(1, 1) = 0.1776356839e-14, (1, 2) = 0., (1, 3) = 0.5329070518e-14, (1, 4) = 0.7105427358e-14, (1, 5) = 0.2220446049e-14, (2, 1) = 0.1776356839e-14, (2, 2) = 0.2664535259e-14, (2, 3) = 0.3552713679e-14, (2, 4) = -0.5551115123e-16, (2, 5) = -0.8881784197e-15, (3, 1) = -0.1065814104e-13, (3, 2) = 0.8881784197e-15, (3, 3) = 0.1332267630e-13, (3, 4) = 0.7993605777e-14, (3, 5) = 0.4884981308e-14, (4, 1) = 0.3552713679e-14, (4, 2) = 0.2220446049e-14, (4, 3) = 0.1065814104e-13, (4, 4) = 0.4440892099e-14, (4, 5) = 0.6217248938e-14, (5, 1) = -0.7993605777e-14, (5, 2) = 0.1776356839e-14, (5, 3) = 0.1598721155e-13, (5, 4) = 0.1243449788e-13, (5, 5) = 0.6217248938e-14, (6, 1) = 0.1065814104e-13, (6, 2) = -0.3330669074e-15, (6, 3) = -0.1776356839e-13, (6, 4) = -0.7105427358e-14, (6, 5) = -0.6217248938e-14})

 (5)

EIGENVALUES

 

> 

S*~S

Vector(6, {(1) = 754.5312606638714, (2) = 512.7138273868854, (3) = 73.24600906942915, (4) = 35.82888224512065, (5) = 4.059820634692874, (6) = 0.})

 (6)

EIGENVECTORS

 

> 

Transpose(Vt)

Matrix(5, 5, {(1, 1) = -.2513827927204978, (1, 2) = .8148366860863387, (1, 3) = -.2606185055842209, (1, 4) = .39672377713059703, (1, 5) = -.21802776368654583, (2, 1) = -.3968455517769299, (2, 2) = .35866150018800186, (2, 3) = .7007682094072526, (2, 4) = -.45071124121664313, (2, 5) = .1402099498711206, (3, 1) = -.6921510074703628, (3, 2) = -.2488880111592855, (3, 3) = -.22081144672043732, (3, 4) = .2513211496937536, (3, 5) = .5891194492399427, (4, 1) = -.3661704447722298, (4, 2) = -.3685935379446182, (4, 3) = .3859384831885419, (4, 4) = .434248601436671, (4, 5) = -.6265282503648171, (5, 1) = -.4076352386533523, (5, 2) = -0.979625692668875e-1, (5, 3) = -.4932501428510237, (5, 4) = -.6226840720358041, (5, 5) = -.4395516923423325})

 (7)

COMPARE LINE (6) AND THE evalues OF  LINE  (8), IN AGREEMENT.

 

COMPARE LINE (7) AND THE evectors OF LINE (8), NOT IN AGREEMENT - differ in sign

 

> 

evalues, evectors:= Eigenvectors(Transpose(X).X)

evalues, evectors := Vector(5, {(1) = 754.531260663872+0.*I, (2) = 512.713827386885+0.*I, (3) = 73.2460090694292+0.*I, (4) = 35.8288822451207+0.*I, (5) = 4.05982063469289+0.*I}), Matrix(5, 5, {(1, 1) = -.251382792720496+0.*I, (1, 2) = -.814836686086340+0.*I, (1, 3) = -.260618505584220+0.*I, (1, 4) = .396723777130597+0.*I, (1, 5) = .218027763686546+0.*I, (2, 1) = -.396845551776929+0.*I, (2, 2) = -.358661500188003+0.*I, (2, 3) = .700768209407252+0.*I, (2, 4) = -.450711241216643+0.*I, (2, 5) = -.140209949871121+0.*I, (3, 1) = -.692151007470364+0.*I, (3, 2) = .248888011159284+0.*I, (3, 3) = -.220811446720437+0.*I, (3, 4) = .251321149693753+0.*I, (3, 5) = -.589119449239943+0.*I, (4, 1) = -.366170444772231+0.*I, (4, 2) = .368593537944617+0.*I, (4, 3) = .385938483188543+0.*I, (4, 4) = .434248601436671+0.*I, (4, 5) = .626528250364817+0.*I, (5, 1) = -.407635238653353+0.*I, (5, 2) = 0.979625692668865e-1+0.*I, (5, 3) = -.493250142851024+0.*I, (5, 4) = -.622684072035804+0.*I, (5, 5) = .439551692342333+0.*I})

 (8)

Find signs of first rows of evectors and Transpose(Vt)

> 

signsev:=convert(map(signum,simplify(fnormal(evectors[1,..]),zero)),list);
signsV :=convert(map(signum,simplify(fnormal(Vt[..,1]),zero)),list);
diffsigns:=zip(`*`,signsev,signsV);

[-1, -1, -1, 1, 1]

  

[-1, 1, -1, 1, -1]

  

[1, -1, 1, 1, -1]

 (9)
> 

Transpose(Vt)-evectors.DiagonalMatrix(diffsigns);

Matrix(5, 5, {(1, 1) = -0.1998401444e-14+0.*I, (1, 2) = -0.8881784197e-15+0.*I, (1, 3) = -0.8326672685e-15+0.*I, (1, 4) = -0.1110223025e-15+0.*I, (1, 5) = -0.8326672685e-16+0.*I, (2, 1) = -0.9436895709e-15+0.*I, (2, 2) = -0.6661338148e-15+0.*I, (2, 3) = 0.5551115123e-15+0.*I, (2, 4) = 0.1665334537e-15+0.*I, (2, 5) = -0.3885780586e-15+0.*I, (3, 1) = 0.1110223025e-14+0.*I, (3, 2) = -0.1221245327e-14+0.*I, (3, 3) = -0.1387778781e-15+0.*I, (3, 4) = 0.5551115123e-15+0.*I, (3, 5) = -0.2220446049e-15+0.*I, (4, 1) = 0.1387778781e-14+0.*I, (4, 2) = -0.1054711873e-14+0.*I, (4, 3) = -0.8881784197e-15+0.*I, (4, 4) = -0.4440892099e-15+0.*I, (4, 5) = -0.2220446049e-15+0.*I, (5, 1) = 0.3330669074e-15+0.*I, (5, 2) = -0.9853229344e-15+0.*I, (5, 3) = 0.5551115123e-15+0.*I, (5, 4) = -0.3330669074e-15+0.*I, (5, 5) = 0.7216449660e-15+0.*I})

 (10)
> 

sdm:= DiagonalMatrix(sqrt~(evalues[1..5]),5,5)

Matrix(5, 5, {(1, 1) = 27.46873241822184+0.*I, (1, 2) = 0.*I, (1, 3) = 0.*I, (1, 4) = 0.*I, (1, 5) = 0.*I, (2, 1) = 0.*I, (2, 2) = 22.64318500977469+0.*I, (2, 3) = 0.*I, (2, 4) = 0.*I, (2, 5) = 0.*I, (3, 1) = 0.*I, (3, 2) = 0.*I, (3, 3) = 8.558388228482581+0.*I, (3, 4) = 0.*I, (3, 5) = 0.*I, (4, 1) = 0.*I, (4, 2) = 0.*I, (4, 3) = 0.*I, (4, 4) = 5.985723201512133+0.*I, (4, 5) = 0.*I, (5, 1) = 0.*I, (5, 2) = 0.*I, (5, 3) = 0.*I, (5, 4) = 0.*I, (5, 5) = 2.0148996587157613+0.*I})

 (11)

THIS SHOULD EQUAL TO THE ORIGINAL X MATRIX

 

> 

U.sdm.Transpose(evectors.DiagonalMatrix(diffsigns));

Matrix(6, 5, {(1, 1) = 8.789999999999969+0.*I, (1, 2) = 9.929999999999986+0.*I, (1, 3) = 9.830000000000018+0.*I, (1, 4) = 5.450000000000025+0.*I, (1, 5) = 3.1600000000000064+0.*I, (2, 1) = 6.10999999999998+0.*I, (2, 2) = 6.909999999999992+0.*I, (2, 3) = 5.040000000000017+0.*I, (2, 4) = -.2699999999999823+0.*I, (2, 5) = 7.980000000000011+0.*I, (3, 1) = -9.150000000000004+0.*I, (3, 2) = -7.930000000000009+0.*I, (3, 3) = 4.859999999999969+0.*I, (3, 4) = 4.849999999999975+0.*I, (3, 5) = 3.0099999999999865+0.*I, (4, 1) = 9.569999999999972+0.*I, (4, 2) = 1.6399999999999917+0.*I, (4, 3) = 8.830000000000009+0.*I, (4, 4) = .7400000000000133+0.*I, (4, 5) = 5.800000000000008+0.*I, (5, 1) = -3.490000000000021+0.*I, (5, 2) = 4.019999999999979+0.*I, (5, 3) = 9.799999999999986+0.*I, (5, 4) = 10.000000000000002+0.*I, (5, 5) = 4.26999999999999+0.*I, (6, 1) = 9.840000000000016+0.*I, (6, 2) = .1500000000000148+0.*I, (6, 3) = -8.98999999999998+0.*I, (6, 4) = -6.019999999999986+0.*I, (6, 5) = -5.309999999999981+0.*I})

 (12)
> 

X - U.sdm.Transpose(evectors.DiagonalMatrix(diffsigns));;

Matrix(6, 5, {(1, 1) = 0.3019806627e-13+0.*I, (1, 2) = 0.1421085472e-13+0.*I, (1, 3) = -0.1776356839e-13+0.*I, (1, 4) = -0.2486899575e-13+0.*I, (1, 5) = -0.6217248938e-14+0.*I, (2, 1) = 0.2042810365e-13+0.*I, (2, 2) = 0.7993605777e-14+0.*I, (2, 3) = -0.1687538997e-13+0.*I, (2, 4) = -0.1770805724e-13+0.*I, (2, 5) = -0.1065814104e-13+0.*I, (3, 1) = 0.3552713679e-14+0.*I, (3, 2) = 0.8881784197e-14+0.*I, (3, 3) = 0.3108624469e-13+0.*I, (3, 4) = 0.2486899575e-13+0.*I, (3, 5) = 0.1332267630e-13+0.*I, (4, 1) = 0.2842170943e-13+0.*I, (4, 2) = 0.8215650382e-14+0.*I, (4, 3) = -0.8881784197e-14+0.*I, (4, 4) = -0.1332267630e-13+0.*I, (4, 5) = -0.7993605777e-14+0.*I, (5, 1) = 0.2087219286e-13+0.*I, (5, 2) = 0.2042810365e-13+0.*I, (5, 3) = 0.1421085472e-13+0.*I, (5, 4) = -0.1776356839e-14+0.*I, (5, 5) = 0.9769962617e-14+0.*I, (6, 1) = -0.1598721155e-13+0.*I, (6, 2) = -0.1479372180e-13+0.*I, (6, 3) = -0.1953992523e-13+0.*I, (6, 4) = -0.1332267630e-13+0.*I, (6, 5) = -0.1865174681e-13+0.*I})

 (13)
> 

 

Download SVD_test2.mw

 

some examples...

dharr 8205
March 08 2024
1 1

Not sure about documentation or the level of explanation you want, but if examples help, here are some:

> 

restart

infix -> prefix

> 

`and`(a, b, c, d)

a and b and c and d

> 

`+`(a, b, c, d)

a+b+c+d

> 

`=`(a, b)

a = b

> 

`union`({a}, {b, c}, {a, d})

{a, b, c, d}

> 

`$`(2, 3)

2, 2, 2

> 

`$`(2, 3)

2, 2, 2

postfix -> prefix

> 

L := [a, b, c, d]

[a, b, c, d]

> 

L[]

a, b, c, d

> 

`?[]`(L)

a, b, c, d

> 

factorial(3)

6

> 

`!`(3); factorial(3)

`!`(3)

6

more...

> 

`and`(L[])

a and b and c and d

> 

(`@`(`and`, `?[]`))(L)

a and b and c and d

> 

`<,>`(a, b)

Vector[column](%id = 36893490135843830708)

> 

`<,>`(a, b)

Vector[column](%id = 36893490135843825652)

> 

`<|>`(a, b)

Vector[row](%id = 36893490135843820236)

> 

`<|>`(a, b)

Vector[row](%id = 36893490135843822756)

> 

NULL

Download prefixes.mw

spacecurve...

dharr 8205
March 07 2024
1 2

A workaround is to use spacecurve instead of plot3d. Unfortunately, the curves need to be done separately.

Perpendicular_3D_lines.mw

3D PDF from Maple 3D plots...

dharr 8205
March 06 2024
2 2

Interesting; I didn't know about such a thing. I looked into it and played around a bit, and various of the exported plot formats can be used to make a u3d file, which can then be imported into Acrobat to make a 3D PDF or used in pdflatex to make a 3D PDF.

Here are the steps I tried

1. Make a 3d plot of a red cube, and export it to a file

p:=plots:-display(plottools:-cuboid([0,0,0],[1,1,1],color=red));
Export("redcube.off", p, base=currentdir);

I tried .off, .ply, .stl via the Export command and .x3d geometry and COLLADA (.dae) via the plot context menu. (These formats are all known to MeshLab.)

2. Convert to .u3d

I used MeshLab from www.meshlab.net (Windows 64 bit but other OSs are available). File -> Import mesh -> choose file. .ply gave an error, .stl asked about removing duplicate vertices, which I accepted. Only .off showed as a red cube; the others had no color.

3a. Open Acrobat Pro and create a blank page .pdf (using file->create). Find the rich media tool and open it. Select add 3d and drag a rectangle where you want the object to be. Choose the .u3d file. You will have to overide a media safety warning, then you can click on the image to activate it. Then click and you can move around. There was no color, and some faces of the cube were transparent. (A sphere from complexplot3d also lost color but was faithfully reproduced.)

3b. The .u3d file can instead be incorporated into .pdf via the LaTeX package media9 and pdfLaTeX. I used overleaf (www.overleaf.com), which uses TeX Live. The .zip file here contains the .tex file, the .u3d file and the generated .pdf file.

3D_PDF.zip

Comments - only the 3D objects are exported, e.g. from POLYGONS PLOT3D structures. These can be colored face by face, but by default the cuboid command sets them all at once to red. I tried coloring different faces differently, but Maple was not exporting the colors to the .obj file (and maybe others).

no value...

dharr 8205
March 06 2024
1 0

The default value is shown when there is no argument passed corresponding to that parameter. (Using NULL achieves the same thing because the NULL is removed from the sequence.)

> 

restart
> 

test := proc (a := 1, b := 2) a+b end proc

proc (a := 1, b := 2) a+b end proc

> 

test(3, 1)

4

> 

test(3)

5

> 

test(3, NULL)

5

> 

3, NULL

3

> 

NULL

Download test.mw

arrows palette...

dharr 8205
March 05 2024
0 0

The arrows palette has diagonal arrows, which can be inserted in 2D, e.g. in a Table:

x

-5

 

-2

 

0

 

2

 

3

 

4

diff(f(x), x)

|

-

0

+

0

-

0

-

0

+

|

f(x)

|

a

lok. min

b

lok. max

a

van v.tg

a

lok. min

b

|
 

Download arrows.mw

 

One way...

dharr 8205
March 05 2024
0 0

Here's one way.

> 

restart
> 

_local(D, gamma)
> 

eqP1 := P1 = (D-C)/(1+gamma); eqP2 := P2 = (D*gamma+C)/(1+gamma)
> 

Theta_exact := -Y*(-D+C)/(1+gamma)+(D*gamma+C)/(1+gamma)

-Y*(-D+C)/(1+gamma)+(D*gamma+C)/(1+gamma)

> 

eliminate({eqP1, eqP2, Theta_exact}, {C, D}); Theta_exact2 := %[2][]

[{C = -P1*gamma+P2, D = P1+P2}, {P1*Y+P2}]

P1*Y+P2

NULL

Download substitution_problem.mw

Mathcontainer...

dharr 8205
March 03 2024
2 1

The font size in a TextArea component cannot be changed as far as I can see. But if you change to a MathContainer component, you can use all the typesetting capabilities. Here is one way to do it:

S4ThalèsAnimation.mw

First 19 20 21 22 23 24 25 Last Page 21 of 81