Matlab Func.

axis

Axis scaling and appearance

Syntax

• axis([xmin xmax ymin ymax])
  axis([xmin xmax ymin ymax zmin zmax cmin cmax])
  v = axis
  
  axis auto
  axis manual
  axis tight
  axis fill
  
  axis ij
  axis xy
  
  axis equal
  axis image
  axis square
  axis vis3d
  axis normal
  
  axis off
  axis on
  

axis(axes_handles,...)

• [mode,visibility,direction] = axis('state')
  

Description

axis manipulates commonly used axes properties. (See Algorithm section.)

axis([xmin xmax ymin ymax]) sets the limits for the x- and y-axis of the current axes.

axis([xmin xmax ymin ymax zmin zmax cmin cmax]) sets the x-, y-, and z-axis limits and the color scaling limits (see caxis) of the current axes.

v = axis returns a row vector containing scaling factors for the x-, y-, and z-axis. v has four or six components depending on whether the current axes is 2-D or 3-D, respectively. The returned values are the current axes' XLim, Ylim, and ZLim properties.

axis auto sets MATLAB to its default behavior of computing the current axes' limits automatically, based on the minimum and maximum values of x, y, and z data. You can restrict this automatic behavior to a specific axis. For example, axis 'auto x' computes only the x-axis limits automatically; axis 'auto yz' computes the y- and z-axis limits automatically.

axis manual and axis(axis) freezes the scaling at the current limits, so that if hold is on, subsequent plots use the same limits. This sets the XLimMode, YLimMode, and ZLimMode properties to manual.

axis tight sets the axis limits to the range of the data.

axis fill sets the axis limits and PlotBoxAspectRatio so that the axes fill the position rectangle. This option has an effect only if PlotBoxAspectRatioMode or DataAspectRatioMode are manual.

axis ij places the coordinate system origin in the upper-left corner. The i-axis is vertical, with values increasing from top to bottom. The j-axis is horizontal with values increasing from left to right.

axis xy draws the graph in the default Cartesian axes format with the coordinate system origin in the lower-left corner. The x-axis is horizontal with values increasing from left to right. The y-axis is vertical with values increasing from bottom to top.

axis equal sets the aspect ratio so that the data units are the same in every direction. The aspect ratio of the x-, y-, and z-axis is adjusted automatically according to the range of data units in the x, y, and z directions.

axis image is the same as axis equal except that the plot box fits tightly around the data.

axis square makes the current axes region square (or cubed when three-dimensional). MATLAB adjusts the x-axis, y-axis, and z-axis so that they have equal lengths and adjusts the increments between data units accordingly.

axis vis3d freezes aspect ratio properties to enable rotation of 3-D objects and overrides stretch-to-fill.

axis normal automatically adjusts the aspect ratio of the axes and the relative scaling of the data units so that the plot fits the figures shape as best as possible.

axis off turns off all axis lines, tick marks, and labels.

axis on turns on all axis lines, tick marks, and labels.

axis(axes_handles,...) applies the axis command to the specified axes. For example, the following statements

• h1 = subplot(221);
  h2 = subplot(222);
  axis([h1 h2],'square')
  

set both axes to square.

[mode,visibility,direction] = axis('state') returns three strings indicating the current setting of axes properties:

Output Argument	Strings Returned
mode	'auto' | 'manual'
visibility	'on' | 'off'
direction	'xy' | 'ij'

mode is auto if XLimMode, YLimMode, and ZLimMode are all set to auto. If XLimMode, YLimMode, or ZLimMode is manual, mode is manual.

Examples

The statements

• x = 0:.025:pi/2;
  plot(x,tan(x),'-ro')
  

use the automatic scaling of the y-axis based on ymax = tan(1.57), which is well over 1000:

The right figure shows a more satisfactory plot after typing

• axis([0  pi/2  0  5])
  
  
  
  

Algorithm

When you specify minimum and maximum values for the x-, y-, and z-axes, axis sets the XLim, Ylim, and ZLim properties for the current axes to the respective minimum and maximum values in the argument list. Additionally, the XLimMode, YLimMode, and ZLimMode properties for the current axes are set to manual.

axis auto sets the current axes' XLimMode, YLimMode, and ZLimMode properties to 'auto'.

axis manual sets the current axes' XLimMode, YLimMode, and ZLimMode properties to 'manual'.

The following table shows the values of the axes properties set by axis equal,
axis normal, axis square, and axis image.

Axes Property	axis equal	axis normal	axis square	axis tightequal
DataAspectRatio	[1 1 1]	not set	not set	[1 1 1]
DataAspectRatioMode	manual	auto	auto	manual
PlotBoxAspectRatio	[3 4 4]	not set	[1 1 1]	auto
PlotBoxAspectRatioMode	manual	auto	manual	auto
Stretch-to-fill	disabled	active	disabled	disabled

balance

Diagonal scaling to improve eigenvalue accuracy

Syntax

• [T,B] = balance(A)
  B = balance(A)
  

Description

[T,B] = balance(A) returns a similarity transformation T such that B = T\A*T, and B has approximately equal row and column norms. T is a permutation of a diagonal matrix whose elements are integer powers of two to prevent the introduction of round-off error. If A is symmetric, then B == A and T is the identity matrix.

B = balance(A) returns just the balanced matrix B.

Remarks

Nonsymmetric matrices can have poorly conditioned eigenvalues. Small perturbations in the matrix, such as roundoff errors, can lead to large perturbations in the eigenvalues. The condition number of the eigenvector matrix,

• cond(V) = norm(V)*norm(inv(V))
  

where

• [V,T] = eig(A)
  

relates the size of the matrix perturbation to the size of the eigenvalue perturbation. Note that the condition number of A itself is irrelevant to the eigenvalue problem.

Balancing is an attempt to concentrate any ill conditioning of the eigenvector matrix into a diagonal scaling. Balancing usually cannot turn a nonsymmetric matrix into a symmetric matrix; it only attempts to make the norm of each row equal to the norm of the corresponding column.

Note The MATLAB eigenvalue function, eig(A), automatically balances A before computing its eigenvalues. Turn off the balancing with eig(A,'nobalance').

Examples

This example shows the basic idea. The matrix A has large elements in the upper right and small elements in the lower left. It is far from being symmetric.

• A = [1  100  10000; .01  1  100; .0001  .01  1]
  A =
     1.0e+04 *
      0.0001    0.0100    1.0000
      0.0000    0.0001    0.0100
      0.0000    0.0000    0.0001
  

Balancing produces a diagonal matrix T with elements that are powers of two and a balanced matrix B that is closer to symmetric than A.

• [T,B] = balance(A)
  T =
     1.0e+03 *
      2.0480         0         0
           0    0.0320         0
           0         0    0.0003
  B =
      1.0000    1.5625    1.2207
      0.6400    1.0000    0.7813
      0.8192    1.2800    1.0000
  

To see the effect on eigenvectors, first compute the eigenvectors of A, shown here as the columns of V.

• [V,E] = eig(A); V
  V =
     -1.0000    0.9999    0.9937
      0.0050    0.0100   -0.1120
      0.0000    0.0001    0.0010
  

Note that all three vectors have the first component the largest. This indicates V is badly conditioned; in fact cond(V) is 8.7766e+003. Next, look at the eigenvectors of B.

• [V,E] = eig(B); V
  V =
     -0.8873    0.6933    0.0898
      0.2839    0.4437   -0.6482
      0.3634    0.5679   -0.7561
  

Now the eigenvectors are well behaved and cond(V) is 1.4421. The ill conditioning is concentrated in the scaling matrix; cond(T) is 8192.

This example is small and not really badly scaled, so the computed eigenvalues of A and B agree within roundoff error; balancing has little effect on the computed results.

Algorithm

balance uses LAPACK routines DGEBAL (real) and ZGEBAL (complex). If you request the output T, it also uses the LAPACK routines DGEBAK (real) and ZGEBAK (complex).

Limitations

Balancing can destroy the properties of certain matrices; use it with some care. If a matrix contains small elements that are due to roundoff error, balancing may scale them up to make them as significant as the other elements of the original matrix.