The Mathematica project will be due on Friday night, 4/27.
Here is the Canon composed at the workshop:
f={1,1,3,3,5,5,1,1,1,1,3,3,5,5,1,1,5,5,6,6,8,8,8,8,5,5,6,6,
8,8,8,8,8,10,8,6,5,5,1,1,8,10,8,6,5,5,1,1,3,3,-4,-4,1,1,1,1};
A=Map[SoundNote,f];
a1=Sound[{"Piano",A},{0,24}];
a2=Sound[{"Flute", RotateRight[A, 8]},{0,24}];
a3=Sound[{"Violin",RotateRight[A,16]},{0,24}];
Sound[{a1,a2,a3},{0,15}]
Here is a "wild piano". It plays a "random matrix". Each
row is an "accord". There are 30 accords. Each tone is chosen
in the range -40,40. We pick a random time (length 1 or 2) to
play each accord.
A = Table[RandomChoice[Range[80] - 40], {30}, {6}];
f[x_] := SoundNote[x, RandomChoice[{1, 2}]];
Sound[{"Piano", Map[f, A]}, {0, 6}]
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The mathematica lab is available here. It is due on
Friday, April 27, 2018. You will submit a notebook with your answers.
We have a workshop on Wednesday, April 25, 2018, 5:45 PM - 6:45 PM in ScienceCtr Hall B.
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There are lots of plotting possibilities to plot systems x' = A x. Here is one
of the simplest. Just enter the following lines into Mathematica and adjust the matrix.
A = {{0, -1}, {-0.4, -1}};
Manipulate[Show[{StreamPlot[A.{x,y}, {x,-5,5},{y,-5,5}],
ParametricPlot[Evaluate[MatrixExp[A t,#] & /@ pt],{t,0,10},
PlotRange->{{-5,5},{-5,5}}]}],{{pt,{{2,0},{0,1},{-3,0}}},
Locator},SaveDefinitions -> True]
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It is useful to know what can be done with computer algebra systems
"the four M's".
Mathematica,
Matlab or
Maple and
Maxima.
The following example snippets should become self explanatory during the course.
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Mathematica
Here are links to get Mathematica
Students
Faculty/Staff.
Make sure to use your Harvard email address when registering.
Contact me (knill@math.harvard.edu) if you plan to use Mathematica on a linux system.
and request a password,
A={{1,2,3},{4,5,5},{6,7,8}}
v={5,-2,3}
Inverse[A]
A.v
A.A.A
LinearSolve[A,v]
RowReduce[A]
QRDecomposition[{{1,0,0},{1,1,0},{1,1,1}}]
Fit[{{0,0},{0,1},{1,3}},{1,x,x^2},x]
CharacteristicPolynomial[A,x]
Tr[A]
Det[A]
Eigenvalues[A]
Eigensystem[A]
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Matlab
Matlab is a CAS which is strong in linear algebra.
Matlab is available as a student version. Here are some
of the above commands in Matlab.
A = [1 2 3; 4 5 5; 6 7 8]
v = [5;-2;3]
inv(A)
A*v
A*A*A
Av
rref(A)
qr(A)
poly(A)
det(A)
trace(A)
eig(A)
[v,d]=eig(A)
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Maple
Maple
is a CAS comparable with Mathematica or Matlab. Here are the same
commands in the Maple dialect.
with(linalg);
A:=[[1,2,3],[4,5,5],[6,7,8]];
v:=[5,-2,3];
inverse(A);
multiply(A,v);
evalm(A*A*A);
linsolve(A,v);
rref(A);
v1:=[1,0,0]; v2:=[1,1,0]; v3:=[1,1,1];
GramSchmidt({v1,v2,v3});
charpoly(A,x);
trace(A);
det(A);
eigenvalues(A);
eigenvectors(A);
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Maxima
Maxima is an open
source CAS originally developed by the
DOE.
While having less features than the commercial CAS, it is
GPL'd
and free software: you can
see the code.
(echelon(A) is here an upper
triangular matrix);
A: matrix([1,2,3],[4,5,5],[6,7,8]);
v: [5,-2,3];
invert(A);
A.v;
A.A.A;
linsolve([x+z=5,x+5*y=-2,x-z=0],[x,y,z]);
echelon(A);
load(eigen); gramschmidt(A);
determinant(A);
charpoly(A,x);
eigenvalues(A);
eigenvectors(A);
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To fit data with Mathematica, you can use
either the built in routines
data={{4,5},{2,10},{1,100},{5,3}};functions={1,x,Sin[x]};
Fit[data,functions,x]
or crank in the linear algebra: a+bx+c sin[x] =y
A=N[{{1,4,Sin[4]},{1,2,Sin[2]},{1,1,Sin[1]},{1,5,Sin[5]}}]; b={5,10,100,3};
Inverse[Transpose[A].A].Transpose[A].b
With both approaches
you get in this example the function 210.3-60x-77.2 Sin[x]
which beset fits the data points (4,5),(2,10),(1,100),(5,3).
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