Differences between revisions 1 and 18 (spanning 17 versions)

Sage Interactions - Linear Algebra

Contents

Sage Interactions - Linear Algebra

Numerical instability of the classical Gram-Schmidt algorithm

by Marshall Hampton (tested by William Stein, who thinks this is really nice!)

xxxxxxxxxx
 
def GS_classic(a_list):
    '''
    Given a list of vectors or a matrix, returns the QR factorization using the classical (and numerically unstable) Gram-Schmidt algorithm.    
    '''
    if type(a_list) != list:
        cols = a_list.cols()
        a_list = [x for x in cols]
    indices = range(len(a_list))
    q = []
    r = [[0 for i in indices] for j in indices]
    v = [a_list[i].copy() for i in indices]
    for i in indices:
        for j in range(0,i):
            r[j][i] = q[j].inner_product(a_list[i])
            v[i] = v[i] - r[j][i]*q[j]
        r[i][i] = (v[i]*v[i])^(1/2)
        q.append(v[i]/r[i][i])
    q = matrix([q[i] for i in indices]).transpose()
    return q, matrix(r)
def GS_modern(a_list):
    '''
    Given a list of vectors or a matrix, returns the QR factorization using the 'modern' Gram-Schmidt algorithm.    
    '''
    if type(a_list) != list:
        cols = a_list.cols()
        a_list = [x for x in cols]
    indices = range(len(a_list))
    q = []
    r = [[0 for i in indices] for j in indices]
    v = [a_list[i].copy() for i in indices]
    for i in indices:
        r[i][i] = v[i].norm(2)
        q.append(v[i]/r[i][i])
        for j in range(i+1, len(indices)):
            r[i][j] = q[i].inner_product(v[j])
            v[j] = v[j] - r[i][j]*q[i]
    q = matrix([q[i] for i in indices]).transpose()
    return q, matrix(r)
html('<h2>Numerical instability of the classical Gram-Schmidt algorithm</h2>')
@interact
def gstest(precision = slider(range(3,53), default = 10), a1 = input_box([1,1/1000,1/1000]), a2 = input_box([1,1/1000,0]), a3 = input_box([1,0,1/1000])):
    myR = RealField(precision)
    displayR = RealField(5)
    html('precision in bits: ' + str(precision) + '<br>')
    A = matrix([a1,a2,a3])
    A = [vector(myR,x) for x in A]
    qn, rn = GS_classic(A)
    qb, rb = GS_modern(A)
    html('Classical Gram-Schmidt:')
    show(matrix(displayR,qn))
    html('Stable Gram-Schmidt:')
    show(matrix(displayR,qb))

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.

xxxxxxxxxx
 
@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)):
    v=vector([r*cos(theta), r*sin(theta)])
    w = A*v
    circles = sum([circle((0,0), radius=i, color='black') for i in [1..2]])
    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)

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.

xxxxxxxxxx
 
from scipy import linalg
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)
    print jsmath('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))

Singular value decomposition

by Marshall Hampton

xxxxxxxxxx
 
import scipy.linalg as lin
var('t')
def rotell(sig,umat,t,offset=0):
    temp = matrix(umat)*matrix(2,1,[sig[0]*cos(t),sig[1]*sin(t)])
    return [offset+temp[0][0],temp[1][0]]
@interact
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)):
    rf_low = RealField(12)
    my_mat = matrix(rf_low,2,2,[a11,a12,a21,a22])
    u,s,vh = lin.svd(my_mat.numpy())
    if ofs: 
        offset = 3
        fsize = 6
        colors = [(1,0,0),(0,0,1),(1,0,0),(0,0,1)]
    else: 
        offset = 0
        fsize = 5
        colors = [(1,0,0),(0,0,1),(.7,.2,0),(0,.3,.7)]
    vvects = sum([arrow([0,0],matrix(vh).row(i),rgbcolor = colors[i]) for i in (0,1)])    
    uvects = Graphics()
    for i in (0,1):
        if s[i] != 0: uvects += arrow([offset,0],vector([offset,0])+matrix(s*u).column(i),rgbcolor = colors[i+2])
    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())))) 
    image_ell = parametric_plot(rotell(s,u,t, offset),(0,2*pi))
    graph_stuff=circle((0,0),1)+image_ell+vvects+uvects
    graph_stuff.set_aspect_ratio(1)
    show(graph_stuff,frame = False,axes=False,figsize=[fsize,fsize])

Discrete Fourier Transform

by Marshall Hampton

xxxxxxxxxx
 
import scipy.fftpack as Fourier
@interact
def discrete_fourier(f = input_box(default=sum([sin(k*x) for k in range(1,5,2)])), scale = slider(.1,20,.1,5)):
    var('x')
    pbegin = -float(pi)*scale
    pend = float(pi)*scale
    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)]
    my_fft = Fourier.fft(f_vals)
    show(list_plot([abs(x) for x in my_fft], plotjoined=True), figsize = [4,3])

The Gauss-Jordan method for inverting a matrix

by Hristo Inouzhe

xxxxxxxxxx
 
#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 = [(M[j,m],j) for j in range(m,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)))

...(goes all the way to invert the matrix)

-  ⇤ ← Revision 1 as of 2008-05-07 13:28:12 → 
  Size: 5314
  Editor: schilly
  Comment:
+   ← Revision 18 as of 2012-04-28 20:09:20 → ⇥
  Size: 10450
  Editor: jason
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 2:
-goto [:interact:interact main page]
+goto [[interact|interact main page]]

<<TableOfContents>>
-Line 6:
+Line 8:
-{{{
+{{{#!sagecell
-Line 60:
+Line 62:
-attachment:GramSchmidt.png
+{{attachment:GramSchmidt.png}}
-Line 68:
+Line 69:
-{{{
@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)):
 Line 74:
-    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]])
    html(""%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
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)
    print jsmath('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}}
-Line 82:
+Line 131:
-{{{
+{{{#!sagecell
-Line 89:
+Line 138:
-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)):
-Line 93:
+Line 142:
-    if ofs == 'On':
+    if ofs:
-Line 106:
+Line 155:
-    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)
+    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))
-Line 110:
+Line 159:
-    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}}
-Line 116:
+Line 164:
-{{{
+{{{#!sagecell
-Line 129:
+Line 177:
-attachment:dfft1.png
+{{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 = [(M[j,m],j) for j in range(m,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(' $M^{- 1} =$ '%latex(M))
    print 'We check it actually is the inverse'
    html(' $M^{- 1} * M =$ '%(latex(M),latex(N),latex(M*N)))
}}}
{{attachment:gauss-jordan.png}}

...(goes all the way to invert the matrix)

Diff for "interact/linear_algebra"