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Solution of a non homogeneous system of linear equations
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goto [:interact:interact main page] | goto [[interact|interact main page]] <<TableOfContents>> |
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by Marshall Hampton (tested by William Stein, who thinks this is really nice!) {{{ |
by Marshall Hampton {{{#!sagecell |
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v = [a_list[i].copy() for i in indices] | v = [a_list[i][:] for i in indices] |
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r[i][i] = (v[i]*v[i])^(1/2) | r[i][i] = (v[i]*v[i])**(1/2) |
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v = [a_list[i].copy() for i in indices] | v = [a_list[i][:] for i in indices] |
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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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attachment:GramSchmidt.png |
{{attachment:GramSchmidt.png}} == Equality of det(A) and det(A.tranpose()) == by Marshall Hampton {{{#!sagecell srg = srange(-4,4,1/10,include_endpoint=True) @interact def dualv(a1=slider(srg,default=1),a2=slider(srg,default=2), a3=slider(srg,default=-1),a4=slider(srg,default=3)): A1 = arrow2d([0,0],[a1,a2],rgbcolor='black') A2 = arrow2d([0,0],[a3,a4],rgbcolor='black') A3 = arrow2d([0,0],[a1,a3],rgbcolor='black') A4 = arrow2d([0,0],[a2,a4],rgbcolor='black') p1 = polygon([[0,0],[a1,a2],[a1+a3,a2+a4],[a3,a4],[0,0]], alpha=.5) p2 = polygon([[0,0],[a1,a3],[a1+a2,a3+a4],[a2,a4],[0,0]],rgbcolor='red', alpha=.5) A = matrix([[a1,a2],[a3,a4]]) 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()))))) show(A1+A2+A3+A4+p1+p2) }}} {{attachment:Det_transpose_new.png}} |
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{{{ @interact def linear_transformation(theta=slider(0, 2*pi, .1), r=slider(0.1, 2, .1, default=1)): A=matrix([[1,-1],[-1,1/2]]) |
{{{#!sagecell @interact def linear_transformation(A=matrix([[1,-1],[-1,1/2]]),theta=slider(0, 2*pi, .1), r=slider(0.1, 2, .1, default=1)): |
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circles = sum([circle((0,0), radius=i, rgbcolor=(0,0,0)) for i in [1..2]]) print jsmath("v = %s,\; %s v=%s"%(v.n(4),latex(A),w.n(4))) show(v.plot(rgbcolor=(1,0,0))+w.plot(rgbcolor=(0,0,1))+circles,aspect_ratio=1) }}} attachment:Linear-Transformations.png |
circles = sum([circle((0,0), radius=i, color='black') for i in [1..2]]) pretty_print(html("$%s %s=%s$"%tuple(map(latex, [A, v.column().n(4), w.column().n(4)])))) show(v.plot(color='red')+w.plot(color='blue')+circles,aspect_ratio=1) }}} {{attachment:Linear-Transformations.png}} == 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. {{{#!sagecell from scipy import linalg pretty_print(html('<h2>The Gerschgorin circle theorem</h2>')) @interact def Gerschgorin(Ain = input_box(default='[[10,1,1/10,0],[-1,9,0,1],[1,0,2,3/10],[-.5,0,-.3,1]]', type = str, label = 'A = '), an_size = slider(1,100,1,1.0)): A = sage_eval(Ain) size = len(A) pretty_print(html('$A = ' + latex(matrix(RealField(10),A))+'$')) A = matrix(RealField(10),A) B = [[0 for i in range(size)] for j in range(size)] for i in range(size): B[i][i] = A[i][i] B = matrix(B) frames = [] centers = [(real(q),imag(q)) for q in [A[i][i] for i in range(size)]] radii_row = [sum([abs(A[i][j]) for j in range(i)+range(i+1,size)]) for i in range(size)] radii_col = [sum([abs(A[j][i]) for j in range(i)+range(i+1,size)]) for i in range(size)] x_min = min([centers[i][0]-radii_row[i] for i in range(size)]+[centers[i][0]-radii_col[i] for i in range(size)]) x_max = max([centers[i][0]+radii_row[i] for i in range(size)]+[centers[i][0]+radii_col[i] for i in range(size)]) y_min = min([centers[i][1]-radii_row[i] for i in range(size)]+[centers[i][1]-radii_col[i] for i in range(size)]) y_max = max([centers[i][1]+radii_row[i] for i in range(size)]+[centers[i][1]+radii_col[i] for i in range(size)]) if an_size > 1: t_range= srange(0,1+1/an_size,1/an_size) else: t_range = [1] for t in t_range: C = t*A + (1-t)*B eigs = [CDF(x) for x in linalg.eigvals(C.numpy())] eigpoints = points([(real(q),imag(q)) for q in eigs],pointsize = 10, rgbcolor = (0,0,0)) centers = [(real(q),imag(q)) for q in [A[i][i] for i in range(size)]] radii_row = [sum([abs(C[i][j]) for j in range(i)+range(i+1,size)]) for i in range(size)] radii_col = [sum([abs(C[j][i]) for j in range(i)+range(i+1,size)]) for i in range(size)] scale = max([(x_max-x_min),(y_max-y_min)]) scale = 7/scale row_circles = sum([circle(centers[i],radii_row[i],fill=True, alpha = .3) for i in range(size)]) col_circles = sum([circle(centers[i],radii_col[i],fill=True, rgbcolor = (1,0,0), alpha = .3) for i in range(size)]) ft = eigpoints+row_circles+col_circles frames.append(ft) show(animate(frames,figsize = [(x_max-x_min)*scale,(y_max-y_min)*scale], xmin = x_min, xmax=x_max, ymin = y_min, ymax = y_max)) }}} {{attachment:Gerschanimate.png}} {{attachment:Gersch.gif}} |
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{{{ | {{{#!sagecell |
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def svd_vis(a11=slider(-1,1,.05,1),a12=slider(-1,1,.05,1),a21=slider(-1,1,.05,0),a22=slider(-1,1,.05,1),ofs= selector(['Off','On'],label='offset image from domain')): | def svd_vis(a11=slider(-1,1,.05,1),a12=slider(-1,1,.05,1),a21=slider(-1,1,.05,0),a22=slider(-1,1,.05,1),ofs= ('offset image from domain',False)): |
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if ofs == 'On': | if ofs: |
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html('<h3>Singular value decomposition: image of the unit circle and the singular vectors</h3>') print jsmath("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())))) image_ell = parametric_plot(rotell(s,u,t, offset),0,2*pi) |
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()))))) image_ell = parametric_plot(rotell(s,u,t, offset),(0,2*pi)) |
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show(graph_stuff,frame = False,axes=False,figsize=[fsize,fsize]) }}} attachment:svd1.png |
show(graph_stuff,frame = False,axes=False,figsize=[fsize,fsize])}}} {{attachment:svd1.png}} |
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{{{ | {{{#!sagecell |
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var('x') | |
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html("<h3>Function plot and its discrete Fourier transform</h3>") show(plot(f, pbegin, pend, plot_points = 512), figsize = [4,3]) f_vals = [f(ind) for ind in srange(pbegin, pend,(pend-pbegin)/512.0)] |
pretty_print(html("<h3>Function plot and its discrete Fourier transform</h3>")) show(plot(f, (x,pbegin, pend), plot_points = 512), figsize = [4,3]) f_vals = [f(x=ind) for ind in srange(pbegin, pend,(pend-pbegin)/512.0)] |
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show(list_plot([abs(x) for x in my_fft], plotjoined=True), figsize = [4,3]) }}} attachment:dfft1.png |
show(list_plot([abs(i) for i in my_fft], plotjoined=True), figsize = [4,3]) }}} {{attachment:dfft1.png}} == The Gauss-Jordan method for inverting a matrix == by Hristo Inouzhe {{{#!sagecell #Choose the size D of the square matrix: D = 3 example = [[1 if k==j else 0 for k in range(D)] for j in range(D)] example[0][-1] = 2 example[-1][0] = 3 @interact def _(M=input_grid(D,D, default = example, label='Matrix to invert', to_value=matrix), tt = text_control('Enter the bits of precision used' ' (only if you entered floating point numbers)'), precision = slider(5,100,5,20), auto_update=False): if det(M)==0: print 'Failure: Matrix is not invertible' return if M.base_ring() == RR: M = M.apply_map(RealField(precision)) N=M M=M.augment(identity_matrix(D)) print 'We construct the augmented matrix' show(M) for m in range(0,D-1): if M[m,m] == 0: lista = [(abs(M[j,m]),j) for j in range(m+1,D)] maxi, c = max(lista) M[c,:],M[m,:]=M[m,:],M[c,:] print 'We permute rows %d and %d'%(m+1,c+1) show(M) for n in range(m+1,D): a=M[m,m] if M[n,m]!=0: print "We add %s times row %d to row %d"%(-M[n,m]/a, m+1, n+1) M=M.with_added_multiple_of_row(n,m,-M[n,m]/a) show(M) for m in range(D-1,-1,-1): for n in range(m-1,-1,-1): a=M[m,m] if M[n,m]!=0: print "We add %s times row %d to the row %d"%(-M[n,m]/a, m+1, n+1) M=M.with_added_multiple_of_row(n,m,-M[n,m]/a) show(M) for m in range(0,D): if M[m,m]!=1: print 'We divide row %d by %s'%(m+1,M[m,m]) M = M.with_row_set_to_multiple_of_row(m,m,1/M[m,m]) show(M) M=M.submatrix(0,D,D) print 'We keep the right submatrix, which contains the inverse' html('$$M^{-1}=%s$$'%latex(M)) print 'We check it actually is the inverse' html('$$M^{-1}*M=%s*%s=%s$$'%(latex(M),latex(N),latex(M*N))) }}} {{attachment:gauss-jordan.png}} ...(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. {{{#!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||width=600}} {{attachment:HSEL_2.png||width=600}} == Solution of a non homogeneous system of linear equations == by Pablo Angulo Coefficients are introduced as a matrix in a single text box, and independent terms as a vector in a separate text box. The number of equations and unknowns are arbitrary. {{{#!sagecell from sage.misc.html import HtmlFragment def SLE_as_latex(A, b, variables): nvars = A.ncols() pretty_print(HtmlFragment( r'$$\left\{\begin{array}{%s}'%('r'*(nvars+1))+ r'\\'.join('%s=&%s'%( (' & '.join((r'%s%s\cdot %s'%('+' if c>0 else '',c,v) if c else '') for c,v in zip(row, variables)) if not row.is_zero() else '&'*(nvars-1)+'0',y) ) for row,y in zip(A,b))+ r'\end{array}\right.$$')) def extended_matrix_as_latex(M): A = M[:,:-1] b = M.column(-1) nvars = A.ncols() pretty_print(HtmlFragment( r'$$\left(\begin{array}{%s}'%('r'*nvars+ '|r')+ r'\\'.join('%s&%s'%( ' & '.join('%s'%c for c in row) ,y) for row,y in zip(A,b))+ r'\end{array}\right)$$')) @interact def SEL(A_text='[(0,0,-1,2),(-1,0,2,4), (0,0,1,-2)]', b_text='[2,1,-2]', auto_update=False ): A = matrix(eval(A_text)) b = vector(eval(b_text)) M = A.augment(b) neqs = len(b) nvars = A.ncols() var_names = ','.join('x%d'%j for j in [1..nvars]) variables = var(var_names) pretty_print(HtmlFragment('Variables: %s'% var_names)) for row,y in zip(A,b): pretty_print(HtmlFragment(sum(c*v for c,v in zip(row, variables))==y)) SLE_as_latex(A, b, variables) pretty_print(HtmlFragment( 'We construct the augmented matrix')) extended_matrix_as_latex(M) pivot = {} ibound_variables = [] for m,row in enumerate(A): 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))) extended_matrix_as_latex(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) extended_matrix_as_latex(M) A = M[:,:-1] b = M.column(-1) SLE_as_latex(A, b, variables) SEL_type = 'compatible' null_rows = None for k,(row,y) in enumerate(zip(A,b)): if row.is_zero(): if y==0 and null_rows is None: null_rows = k break elif y!=0: SEL_type = 'incompatible' if SEL_type == 'incompatible': pretty_print( HtmlFragment('The system has no solutions')) return if null_rows: pretty_print(HtmlFragment('We remove trivial 0=0 equations')) A = A[:null_rows,:] b = b[:null_rows] SLE_as_latex(A, b, 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))==y for row,y in zip(A,b) ] 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('Solution in vector form')) pretty_print( HtmlFragment( r'$$ %s + \left\langle %s\right\rangle$$'%( latex(parametric_sol.subs(dict(zip(free_variables, [0]*len(free_variables))))), ','.join(latex( parametric_sol.apply_map(lambda s:s.diff(v)) ) for v in free_variables) if free_variables else latex(matrix([0]*nvars).transpose())) )) pretty_print( HtmlFragment('Dimension is %d'%len(free_variables))) }}} {{attachment:NHSEL_1.png||width=600}} {{attachment:NHSEL_2.png||width=600}} |
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
- Solution of a non 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.
Solution of a non homogeneous system of linear equations
by Pablo Angulo
Coefficients are introduced as a matrix in a single text box, and independent terms as a vector in a separate text box. The number of equations and unknowns are arbitrary.