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Comment: Fix suggested by Sharon Robbert
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attachment:HSEL_1.png
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Deletions are marked like this. | Additions are marked like this. |
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html('<h2>Numerical instability of the classical Gram-Schmidt algorithm</h2>') | pretty_print(html('<h2>Numerical instability of the classical Gram-Schmidt algorithm</h2>')) |
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html('precision in bits: ' + str(precision) + '<br>') | pretty_print(html('precision in bits: ' + str(precision) + '<br>')) |
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html('Classical Gram-Schmidt:') | pretty_print(html('Classical Gram-Schmidt:')) |
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html('Stable Gram-Schmidt:') | pretty_print(html('Stable Gram-Schmidt:')) |
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html('<h3>The determinant of a matrix is equal to the determinant of the transpose</h3>') html("$det(%s) = det(%s)=%s$"%(latex(A),latex(A.transpose()),latex(RR(A.determinant())))) |
pretty_print(html('<h3>The determinant of a matrix is equal to the determinant of the transpose</h3>')) pretty_print(html("$\det(%s) = \det(%s)=%s$"%(latex(A),latex(A.transpose()),latex(RR(A.determinant()))))) |
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{{attachment:Det_transpose.png}} | {{attachment:Det_transpose_new.png}} |
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html("$%s %s=%s$"%tuple(map(latex, [A, v.column().n(4), w.column().n(4)]))) | pretty_print(html("$%s %s=%s$"%tuple(map(latex, [A, v.column().n(4), w.column().n(4)])))) |
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html('<h2>The Gerschgorin circle theorem</h2>') | pretty_print(html('<h2>The Gerschgorin circle theorem</h2>')) |
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html('$A = ' + latex(matrix(RealField(10),A))+'$') | pretty_print(html('$A = ' + latex(matrix(RealField(10),A))+'$')) |
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html('<h3>Singular value decomposition: image of the unit circle and the singular vectors</h3>') html("$A = %s = %s %s %s$"%(latex(my_mat), latex(matrix(rf_low,u.tolist())), latex(matrix(rf_low,2,2,[s[0],0,0,s[1]])), latex(matrix(rf_low,vh.tolist())))) |
pretty_print(html('<h3>Singular value decomposition: image of the unit circle and the singular vectors</h3>')) pretty_print(html("$A = %s = %s %s %s$"%(latex(my_mat), latex(matrix(rf_low,u.tolist())), latex(matrix(rf_low,2,2,[s[0],0,0,s[1]])), latex(matrix(rf_low,vh.tolist()))))) |
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html("<h3>Function plot and its discrete Fourier transform</h3>") | pretty_print(html("<h3>Function plot and its discrete Fourier transform</h3>")) |
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lista = [(M[j,m],j) for j in range(m+1,D)] | lista = [(abs(M[j,m]),j) for j in range(m+1,D)] |
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== Solution of an homogeneous system of linear equations == by Pablo Angulo Coefficients are introduced as a matrix in a single text box. The number of equations and unknowns are arbitrary. {{{#!sagecell from sage.misc.html import HtmlFragment def HSLE_as_latex(A, variables): nvars = A.ncols() pretty_print(HtmlFragment( r'$$\left\{\begin{array}{%s}'%('r'*(nvars+1))+ r'\\'.join('%s=&0'%( ' & '.join((r'%s%s\cdot %s'%('+' if c>0 else '',c,v) if c else '') for c,v in zip(row, variables)) if not row.is_zero() else '&'*(nvars-1)+'0' ) for row in A)+ r'\end{array}\right.$$')) @interact def SEL(A='[(0,1,-1,2),(-1,0,2,4), (0,-1,1,-2)]', auto_update=False ): M = A = matrix(eval(A)) neqs = M.nrows() nvars = M.ncols() var_names = ','.join('x%d'%j for j in [1..nvars]) variables = var(var_names) HSLE_as_latex(M, variables) pretty_print(HtmlFragment( 'SEL in matrix form')) show(M) pivot = {} ibound_variables = [] for m,row in enumerate(M): for k in range(m,nvars): lista = [(abs(M[j,k]),j) for j in range(m,neqs)] maxi, c = max(lista) if maxi > 0: ibound_variables.append(k) if M[m,k]==0: M[c,:],M[m,:]=M[m,:],M[c,:] pretty_print( HtmlFragment('We permute rows %d and %d'%(m+1,c+1))) show(M) pivot[m] = k break a=M[m,k] for n in range(m+1,neqs): if M[n,k]!=0: pretty_print( HtmlFragment("We add %s times row %d to row %d"%(-M[n,k]/a, m+1, n+1))) M=M.with_added_multiple_of_row(n,m,-M[n,k]/a) show(M) pretty_print( HtmlFragment('Equivalent system of equations')) HSLE_as_latex(M, variables) SEL_type = 'compatible' null_rows = None for k,row in enumerate(M): if row.is_zero(): pretty_print( HtmlFragment('We remove trivial 0=0 equations')) M = M[:k,:] HSLE_as_latex(M, variables) ifree_variables = [k for k in range(nvars) if k not in ibound_variables] bound_variables = [variables[k] for k in ibound_variables] free_variables = [variables[k] for k in ifree_variables] pretty_print( HtmlFragment('Free variables: %s'% free_variables)) pretty_print( HtmlFragment('Bound variables: %s'% bound_variables)) reduced_eqs = [ sum(c*v for c,v in zip(row, variables))==0 for row in M ] xvector = vector(variables) if len(bound_variables)==1: soldict = solve(reduced_eqs, bound_variables[0], solution_dict=True)[0] else: soldict = solve(reduced_eqs, bound_variables, solution_dict=True)[0] pretty_print( HtmlFragment('Solution in parametric form')) parametric_sol = matrix( xvector.apply_map(lambda s:s.subs(soldict)) ).transpose() show(parametric_sol) pretty_print( HtmlFragment('Generators')) pretty_print( HtmlFragment( r'$$\langle %s\rangle$$'%','.join(latex( parametric_sol.subs(dict((variables[k],1 if j==k else 0) for k in ifree_variables)) ) for j in ifree_variables) )) pretty_print( HtmlFragment('Dimension is %d'%len(free_variables))) }}} {{attachment:HSEL_1.png}} {{attachment:HSEL_2.png}} |
Sage Interactions - Linear Algebra
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Contents
-
Sage Interactions - Linear Algebra
- Numerical instability of the classical Gram-Schmidt algorithm
- Equality of det(A) and det(A.tranpose())
- Linear transformations
- Gerschgorin Circle Theorem
- Singular value decomposition
- Discrete Fourier Transform
- The Gauss-Jordan method for inverting a matrix
- Solution of an homogeneous system of linear equations
Numerical instability of the classical Gram-Schmidt algorithm
by Marshall Hampton
Equality of det(A) and det(A.tranpose())
by Marshall Hampton
Linear transformations
by Jason Grout
A square matrix defines a linear transformation which rotates and/or scales vectors. In the interact command below, the red vector represents the original vector (v) and the blue vector represents the image w under the linear transformation. You can change the angle and length of v by changing theta and r.
Gerschgorin Circle Theorem
by Marshall Hampton. This animated version requires convert (imagemagick) to be installed, but it can easily be modified to a static version. The animation illustrates the idea behind the stronger version of Gerschgorin's theorem, which says that if the disks around the eigenvalues are disjoint then there is one eigenvalue per disk. The proof is by continuity of the eigenvalues under a homotopy to a diagonal matrix.
Singular value decomposition
by Marshall Hampton
Discrete Fourier Transform
by Marshall Hampton
The Gauss-Jordan method for inverting a matrix
by Hristo Inouzhe
...(goes all the way to invert the matrix)
Solution of an homogeneous system of linear equations
by Pablo Angulo
Coefficients are introduced as a matrix in a single text box. The number of equations and unknowns are arbitrary.