Mathematica Primer.nb 1

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Horace Greene
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1 Mathematica Primer.nb (* Mathematica is a powerful mathematics software pacage that can perform numerical calculations and SYMBOIC manipulations. To enter the program from either a Mac or Windows environment, open or double clic the Mathematica icon: After seeing colorful graphics, a prompt will appear. *) (* Note: parenthesis + asteris => comment field which is ignored during execution of statements. *) (* Mathematica can perform simple arithmetic. Add two numbers: type: <enter> or <shift><return> to execute: (<return> used for typing) *) (* Result displayed. Note comment field was ignored. *) (* We can also Subtract: *) 3-2 (* Divide: *) 5/3 5 (* and multiply: *) 33*8 264 (* Note: a space can replace the * when multiplying: *) (* Pi represents the number p, stored symbolically in Mathematica. The notation //N gives a 6-digit numerical approximation: *) Pi//N N[Pi,77] (* approximates p to 77 digits! N[f,# digits] => approximate f to # digits*) Ö Ö 3852

2 Mathematica Primer.nb 2 (* Special functions such as sine are denoted by a capital letter and a square bracet: *) Sin[Pi/2] (* Special functions are listed in the Appendix of the Mathematica reference text. *) (* We can define functions such as polynomials, which Mathematica can manipulate symbolically, just lie we do: *) F = a + b x + c x^2 + d x^3 a + b x + c x 2 + d x 3 (* Mathematica can also differentiate: *) F = D[F,x] b + 2 c x + 3 d x 2 (* Multiple derivatives: *) F3 = D[F, {x,3}] 6 d (* Partial derivatives: *) Fxc = D[F,x,c] 2 x (* Mathematica can solve indefinite integrals: *) Fintegral = Integrate[F,x] a x + b x2 ÅÅÅÅÅÅÅÅÅ 2 + ÅÅÅÅÅÅÅÅÅ c x3 3 + ÅÅÅÅÅÅÅÅÅ d x4 4 (* or definte integrals: *) Integrate[F, {x,,xmax}] b c d b xmax c xmax d xmax -a a xmax (* Graphs: suppose we let *) a = 3; b = ; c = ; d = ; (* Note the semicolon permits sequential execution needed in a program, but suppresses display of the result. *)

3 Mathematica Primer.nb 3 (* and then plot F over the domain - x 3 *) Plot[F, {x, -, 3}] (* We can remove the value assigned to a variable: *) a =.; (* and change the assignment: *) a = 30 Cos[Pi y^2]; (* Now let a = a(y) be a second variable for a 3D plot: *) Plot3D[F, {x, -, 3}, {y,, 4}] SurfaceGraphics- (* et's chec what is currently assigned to F: *) F x + x + x + 30 Cos[Pi y ]

4 Mathematica Primer.nb 4 (* We can plot more extensive functions: *) Plot[ Sum[2/(i*Pi)*(-^(i+))*Sin[i*Pi*x], {i,,0}], {x,0,}, Axesabel->{"x","y"}] y x (* We can expand expressions: *) Gp = Expand[(x+)^3] x + 3 x + x (* and evaluate the expression, for say x = 0: *) Gp/.x->0 (* or for x = r + s: *) Gp/.x->(r+s) (r + s) + 3 (r + s) + (r + s) (* We can simplify expressions, although Mathematica lacs intuition: *) Simplify[u^4*x + 4*u^3*x^2 + 6*u^2*x^3 + 4*u*x^4 + x^5 - u^4*z - 4*u^3*x*z - 6*u^2*x^2*z - 4*u*x^3*z - x^4*z] 4 (u + x) (x - z) (* Mathematica can perform matrix operations, and has the needed Do loops, If statements, Print, etc. that facilitate sequential programming: *) Q = Array[q, {4,4}]; (* defines a 4x4 array Q *) Do[ Q[[i,j]] = i^(j-), {i,,4},{j,,4}]; (* assigns values to array Q *) Q (* will display the array values row by row: *) {{,,, }, {, 2, 4, 8}, {, 3, 9, 27}, {, 4, 6, 64}}

5 Mathematica Primer.nb 5 (* Get a specific element *) Q[[2,4]] 8 (* We can define matrices directly, and even let elements be symbolic variables: *) A = {{ R/, 0, -Kxc/, 0}, { KeKh, -we, 0, 0}, { 0, Kf w^2/, -2 z w, -w^2 + Kxp Kf w^2/}, { 0, 0,, 0}} (* Note: column elements aligned for programmer recognition. *) 99 ÅÅÅÅ R Kxc Kf w2, 0, - ÅÅÅÅÅÅÅÅÅ, 0=, 8KeKh, -we, 0, 0<, 90, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ, -2 w z, -w 2 Kf Kxp w2 + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ =, 80, 0,, 0<= (* Tae determinant *) DETA = Det[A] - R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxp R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ (* Multiply matrices, using period or. as the operator: *) A.Q Kxc R -3 Kxc R -9 Kxc R -27 Kxc R {{-(---) , , , }, {KeKh - we, KeKh - 2 we, KeKh - 4 we, KeKh - 8 we}, Kf w Kf Kxp w {-w w z, Kf w 2 Kf Kxp w (-w ) - 6 w z, Kf w 2 Kf Kxp w (-w ) - 8 w z, Kf w 2 Kf Kxp w (-w ) - 54 w z}, {, 3, 9, 27}} (* and vectors: *) Y = Q.{,2,3,4} {0, 49, 42, 33} (* Select a vector component *) Y[[3]] 42

6 Mathematica Primer.nb 6 (* and even tae the symbolic inverse! *) Inverse[A] 99 -w2 we + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Kf Kxp w2 we ÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ, 0, 0, - Kxc w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxc Kxp w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ =, - R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxp R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxp R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ 9 -KeKh w2 KeKh Kf Kxp w2 + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ, - R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxp R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ R w ÅÅÅÅÅÅÅÅÅÅ 2 Kf Kxp R w2 - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxp R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ, 0, KeKh Kxc w2 KeKh Kf Kxc Kxp w2 - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxp R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ KeKh Kf w 80, 0, 0, <, 9- ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 ÅÅÅÅÅÅÅÅÅÅÅÅ I- R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxp R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ M, 2 Kf R w ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ I- R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxp R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ M, R we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ I- R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxp R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ M, - 2 R w we z ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ == - R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅ + Kf Kxp R w2 we ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ KeKh Kf Kxc w 2 (* We can assign values to elements of A: *) R = 0.36; = ^-3; w = 29500; z = 0.26; = ^7; we = 02; e = ^-7; h = 5249; KeKh = e h; Kxc = 50; Kxp = 0.23; Kf = ^7; (* and then inspect the matrix. *) A//N {{54.76, 0, , 0}, { , -02., 0, 0}, 8 8 {0, , , }, {0, 0,., 0}} (* Mathematica can also get eigenvalues and eigenvectors: *) Eigensystem[A] ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ =, {{334.9, , I, I}, {{ , , , }, { , , , }, { I, I, I, I}, { I, I, I, I}}} (* Here I = Sqrt[-] is the base of imaginary numbers *)

7 Mathematica Primer.nb 7 (* Many special functions are available. Suppose we plot multiple curves on the same graph: *) Plot[ {BesselJ[0,x], BesselJ[,x]}, {x,0,0}] (* Parametric plots--plots where the absisca and ordinate depend on a third independent variable--permit Bode plots: *) (* Bode magnitude plot for *) n = 5; (* Butterworth filter: *) ParametricPlot[ {og[0,w],20 og[0,/sqrt[+w^(2 n)]]}, {w,0^-2,0^2},plotpoints->200,plotrange->all, AxesOrigin->{-2,0}]

8 Mathematica Primer.nb 8 (* Bode phase plot for same Butterworth filter: *) ParametricPlot[{og[0,w],-80/Pi (ArcTan[,w] + ArcTan[ - w^2, *w ] + ArcTan[ - w^2, *w ])}, {w,0^-2,0^2}, PlotPoints->200,PlotRange->All,AxesOrigin->{2,-450}] (* Mathematica can find roots of polynomials. Note that the double equal sign == denotes an equation, rather than an assignment: *) NSolve[ x^5 + 6 x^4 + x^3-2 x^2 + x - 5 == 0, x] {{x -> }, {x -> }, {x -> I}, {x -> I}, {x -> }} (* Roots of other transcendental equations: *) FindRoot[ {Sin[x] + 4 Cos[2 x] == y, 2 x == y}, {x, 0},{y,0}] {x -> , y ->.38332} (* Mathematica can generate closed form SYMBOIC solutions to simple (especially linear) differential equations: *) DSolve[ { x'[t] == - x[t] + z[t], z''[t] + 2 z'[t] + z[t] == Sin[2 t], (* differential equations *) x[0] == ao, z[0] == 6, z'[0] == -20 (* initial conditions, one is SYMBOIC *) }, {x[t], z[t]},t] (* Note: prime denotes derivative term(s). Time dependence in differential equations expressed as x[t] and z[t]. Unnowns are {x[t], z[t]}. *) 2 t t ao t t + 2 E Cos[2 t] - E Sin[2 t] {{x[t] -> , t 25 E t t t - 4 E Cos[2 t] - 3 E Sin[2 t] z[t] -> }} t 25 E

9 Mathematica Primer.nb 9 (* Mathematica can also generate numerical solutions to differential equations: *) X = NDSolve[ { x[]'[t] == -x[][t] + z[t], x[2]'[t] == x[][t] - x[2][t], z''[t] + 2 z'[t] + z[t] == Sin[2 t], (* differential equations *) x[][0] ==, x[2][0] == 0, z[0] == 6, z'[0] == -20 (* initial conditions *) }, {x[],x[2],z}, {t,0,}] (* Note: Prime denotes derivative term(s). Variable x is now a vector with two components x[] and x[2], with time dependence in differential equations expressed as x[][t] and x[2][t]. Solution for unnowns {x[],x[2],z} is stored in X. *) 88x@D Ø InterpolatingFunction@880.,.<<, <>D, x@2d Ø InterpolatingFunction@880.,.<<, <>D, z Ø InterpolatingFunction@880.,.<<, <>D<< Plot[ Evaluate[ x[][t]/.x ], {t,0,}] Ü Graphics Ü Plot[Evaluate[ x[2][t]/.x ], {t,0,}] Ü Graphics Ü

10 Mathematica Primer.nb 0 Plot[Evaluate[ z[t]/.x ], {t,0,}] Ü Graphics Ü H* Program for producing Bode Plots Transfer function: G Hs * G = 50 Hs - 4 ê HH0 s + Hs^2 + 4 s H-4 + s ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ H + 0 s H8 + 4 s + s 2 Solve@Numerator@GD == 0, sd 88s Ø 4<< Solve@Denominator@GD == 0, sd 98s Ø -2-2 I<, 8s Ø I<, 9s Ø - ÅÅÅÅÅÅÅ 0 == Mag = Abs@G ê. Hs -> I wd; Phase = Arg@G ê. Hs -> I wd; ParametricPlot@88 og@0, wd, 20 og@0, MagD <, 8 og@0, wd, Phase 80 ê Pi <<, 8w, 0^H-3, 0^H3.5<, PlotRange -> All, PlotPoints -> 500, Plotabel -> "Magnitude HdB & Phase Hdegree", Axes -> None, Frame -> TrueD Magnitude HdB & Phase Hdegree Ü Graphics Ü

11 Mathematica Primer.nb (* Appendix: Mathematica program for calculating transfer function and determinants, for band pass filter circuit. *) (* System matrix *) A = { { 0, /, 0, 0, 0, 0}, {-/C, 0, -/C2, 0, 0, 0}, { 0, /, 0, -/2, 0, -/}, { 0, 0, /C2, 0, 0, 0}, { 0, 0, 0, 0, 0, /}, { 0, 0, /C2, 0, -/C,-Rload/}} 990, ÅÅÅÅÅÅÅ, 0, 0, 0, 0=, 9- ÅÅÅÅÅÅÅ C, 0, - ÅÅÅÅÅÅÅ C2 90, 0, ÅÅÅÅÅÅÅ, 0, 0, 0=, 90, 0, 0, 0, 0, ÅÅÅÅÅÅÅ C2 H* Calculate determinant * determinanta = Det@AD ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ C 2 C2 2 2 (* Input vector *) b = { 0, Ein, 0, 0, 0, 0} {0, Ein, 0, 0, 0, 0} (* Matrix: [si - A] *) As = s IdentityMatrix[6] - A, 0, 0, 0=, 90, ÅÅÅÅÅÅÅ =, 90, 0, ÅÅÅÅÅÅÅ {{s, -(--), - 0, 0, 0, 0}, {--, - s, --, - 0, 0, 0}, C C2 {0, -(--), - s, --, - 0, --}, - {0, 0, -(--), - s, 0, 0}, 2 C2 Rload {0, 0, 0, 0, s, -(--)}, - {0, 0, -(--), - 0, --, s}} C2 C, 0, - ÅÅÅÅÅÅÅ 2, 0, - ÅÅÅÅÅÅÅ =, C2, 0, - ÅÅÅÅÅÅÅ C, - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Rload == AT = Transpose[As]; (* Transpose As: exchange rows/columns *) AT[[6]] = b; (* Replace 6th column with b *) AT (* Inspect results *) {{s, --, - 0, 0, 0, 0}, {-(--), - s, -(--), - 0, 0, 0}, C {0, --, - s, -(--), - 0, -(--)}, - {0, 0, --, - s, 0, 0}, C2 C2 C2 2 {0, 0, 0, 0, s, --}, - {0, Ein, 0, 0, 0, 0}} C

12 Mathematica Primer.nb 2 (* Transfer function via Cramers rule *) Eout = Rload/ (Det[AT]/Det[As]) 2 3 (C Ein 2 Rload s ) / ( + C Rload s + 2 C s + 2 C 2 s + C2 2 s C Rload s + C 2 Rload s + C C2 2 Rload s C s + 2 C 2 s + 2 C C2 2 s C C2 2 Rload s + C C2 2 s ) G = Eout/Ein 2 3 (C 2 Rload s ) / ( + C Rload s + 2 C s + 2 C 2 s + C2 2 s C Rload s + C 2 Rload s + C C2 2 Rload s C s + 2 C 2 s + 2 C C2 2 s C C2 2 Rload s + C C2 2 s ) (* Transfer function via Matrix Inverse: all transfer functions calculated simultaneously. *) AInverse = Inverse[As].b; (* Calculate; A Inverse b *) Gi = (Rload/ AInverse[[6]])/Ein (* Display only G(s): *) 2 3 (C 2 Rload s ) / ( + C Rload s + 2 C s + 2 C 2 s + C2 2 s C Rload s + C 2 Rload s + C C2 2 Rload s C s + 2 C 2 s + 2 C C2 2 s C C2 2 Rload s + C C2 2 s ) (* Insert parameter values into transfer function: *) = 0^-3; 2 = 0^-3; C = 0^-6; C2 = 0^-6; Rload = 00; G//N (* Inspect transfer function *) -3 3 (. 0 s ) / ( s s s s s +. 0 s )

13 Mathematica Primer.nb 3 GPlot = (G/.s->(I w))//n -3 3 (-. 0 I w ) / ( I w w I w w I w -. 0 w ) (* Bode magnitude plot *) ParametricPlot[{og[0,w],20 og[0, Abs[GPlot]]}, {w,0^3,0^6}, PlotRange->All, PlotPoints->500, Plotabel->"Magnitude (db)", Axes->None, Frame->True] Magnitude (db) (* Bode phase plot *) ParametricPlot[{og[0,w],80/Pi Arg[GPlot]}, {w,0^3,0^6}, PlotRange->All, PlotPoints->500, Plotabel->"Phase (degrees)", Axes->None, Axesabel->{"w","Phase"}, Frame->True] Phase (degrees)

14 Mathematica Primer.nb 4 (* Bode magnitude & phase plot *) ParametricPlot[{{og[0,w],20 og[0, Abs[GPlot]]}, {og[0,w],80/pi Arg[GPlot]}}, {w,0^3,0^6}, PlotRange->All, PlotPoints->500, Plotabel->"Magnitude(dB) & Phase(degree)", Axes->None, Frame->True] Magnitude(dB) & Phase(degree) (* Insert parameter values into transfer function: *) =.; 2 =.; C = 0^-6; C2 = 0^-6; Rload = 00; G//N (* Inspect transfer function *) AT = Transpose[As]; (* Transpose As: exchange rows/columns *) AT[[2]] = b; (* Replace 6th column with b *) AT (* Inspect results *) {{s, , 0, 0, 0, 0}, {0, Ein, 0, 0, 0, 0}, {0, , s, , 0, }, {0, 0, 000, s, 0, 0}, {0, 0, 0, 0, s, }, {0, 0, 000, 0, -000, s}}

15 Mathematica Primer.nb 5 (* Transfer function for input impedance via Cramers rule *) λ = Simplify[ Ein/(Det[AT]/Det[As])]//N (0.00 ( s s s s s + s )) / (s ( s s s + s )) (* Other commands and examples can be found in Mathematica, A System for Doing Mathematics by Computer Steven Wolfram Addison-Wesley *)

Solving Schrödinger equation numerically

Solving Schrödinger equation numerically eigenvalues.nb Solving Schödinge equation numeically Basic idea on woking out the enegy eigenvalues numeically is vey simple. Just solve the Schödinge equation with a guessed enegy, and it always makes