#
#
# WORKSHEET#16
#
# Classification of critical points
#
#
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# In this worksheet we continue the study of the critical points of the the function f(x,y)
# from worksheet#15. This is done through the hessian matrix.
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> f:=(x,y)->sin(y-x^2-1)+cos(2*y^2-x);
2 2
f := (x,y) -> sin(y - x - 1) + cos(2 y - x)
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> f1:=D[1](f);f2:=D[2](f);
2 2
f1 := (x,y) -> - 2 cos(- y + x + 1) x - sin(- 2 y + x)
2 2
f2 := (x,y) -> cos(- y + x + 1) + 4 sin(- 2 y + x) y
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> with(linalg);
Warning: new definition for norm
Warning: new definition for trace
[BlockDiagonal, GramSchmidt, JordanBlock, Wronskian, add, addcol, addrow, adj,
adjoint, angle, augment, backsub, band, basis, bezout, blockmatrix,
charmat, charpoly, col, coldim, colspace, colspan, companion, concat, cond,
copyinto, crossprod, curl, definite, delcols, delrows, det, diag, diverge,
dotprod, eigenvals, eigenvects, entermatrix, equal, exponential, extend,
ffgausselim, fibonacci, frobenius, gausselim, gaussjord, genmatrix, grad,
hadamard, hermite, hessian, hilbert, htranspose, ihermite, indexfunc,
innerprod, intbasis, inverse, ismith, iszero, jacobian, jordan, kernel,
laplacian, leastsqrs, linsolve, matrix, minor, minpoly, mulcol, mulrow,
multiply, norm, normalize, nullspace, orthog, permanent, pivot, potential,
randmatrix, randvector, rank, ratform, row, rowdim, rowspace, rowspan,
rref, scalarmul, singularvals, smith, stack, submatrix, subvector,
sumbasis, swapcol, swaprow, sylvester, toeplitz, trace, transpose,
vandermonde, vecpotent, vectdim, vector]
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> ?hessian;
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> H:=hessian(f(x,y),[x,y]);
H := 2 2 2 2
[4 sin(- y + x + 1) x - 2 cos(- y + x + 1) - cos(- 2 y + x),
2 2
- 2 sin(- y + x + 1) x + 4 cos(- 2 y + x) y]
2 2
[- 2 sin(- y + x + 1) x + 4 cos(- 2 y + x) y,
2 2 2 2
sin(- y + x + 1) - 16 cos(- 2 y + x) y + 4 sin(- 2 y + x)]
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# Note that this matrix is symmetric. A critical point will be a local mimimum if all the
# eigenvalues of the hessian, when evaluated at the critical point, are positive and a local
# maximum if they are all negative. If all eigenvalues at the critical point are non-zero,but
# some are negative and some are positive, then the critical point is a saddle point. This
# works for all dimensions.
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> ?eigenvalues
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> fsolve ({f1(x,y),f2(x,y)},{x,y},{x=0.36..0.38,y=-0.44..-0.42});
{x = .3728572391, y = -.4317738060}
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> assign(");
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> print(x,y,f1(x,y),f2(x,y),f(x,y));
-10 -9
.3728572391, -.4317738060, .529485605*10 , -.3778129031*10 , 0
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> print(H(x,y));
2
[4 %2 x(.3728572391, -.4317738060)
2
- 2 cos(- y + x + 1)(.3728572391, -.4317738060) - %1,
- .7457144782 %2 - 1.727095224 %1]
2
[ - .7457144782 %2 - 1.727095224 %1, %2 - 16 %1 y(.3728572391, -.4317738060)
2
+ 4 sin(- 2 y + x)(.3728572391, -.4317738060)]
2
%1 := cos(- 2 y + x)(.3728572391, -.4317738060)
2
%2 := sin(- y + x + 1)(.3728572391, -.4317738060)
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#
> x;y;
.3728572391
-.4317738060
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# Thus even though x and y have been assigned values H has not. H is a 2 by 2 matrix
# whose entries are H[1,1], H[1,2], etc. The entries have been assigned values.
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> H[1,1];
-.4439099168
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> ?matrix
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> A:=matrix(2,2,[H[1,1],H[1,2],H[2,1],H[2,2]]);
[ -.4439099168 -2.472809702 ]
A := [ ]
[ -2.472809702 -1.982857912 ]
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> eigenvals(A);
-3.803148007, 1.376380179
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# Thus this crirical point is a saddle point, that is the in some directions near the critical
# point the function is increasing and in others it is decreasing. We'll try one more critical
# point.
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> x:='x':y:='y':
> fsolve({f1(x,y),f2(x,y)},{x,y},{x=-1.4..-1.0,y=0.8..1.2});
{y = .9743122425, x = -1.243023962}
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> assign(");
--------------------------------------------------------------------------------
> print(x,y,f1(x,y),f2(x,y),f(x,y));
-8 -9
-1.243023962, .9743122425, -.3406151520*10 , .393574498*10 , -2.
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> A:=matrix(2,2,[H[1,1],H[1,2],H[2,1],H[2,2]]);
[ 7.180434282 -1.411201046 ]
A := [ ]
[ -1.411201046 16.18854953 ]
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> eigenvals(");
6.964531797, 16.40445202
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# Both eigenvalues are positive so this is a local minimum. Actually this is a global
# minimum. To see this note that the value of f(x,y) is -2 and then just recall the definition
# of f(x,y).
