MATLAB: What is the difference between Gram-Smith QR decomposition procedure and qr.m function in Matlab

linear algebraMATLABqr decompositionqr factorization

Hi everyone!

I want to know what is the differnce between the Gram-Smith procedure and qr.m function in matlab. Why there is a fourth coulmn resulting from using qr.m function? The photo is attached for the Gram-Smith procedure .Any help will be so appreciated.

The code is
A=[1 -2 -1; 2 0 1; 2 -4 2;4 0 0];
[Q,R] = qr(A)
The result is
Q =
   -0.2000    0.4000    0.8000   -0.4000
   -0.4000   -0.2000   -0.4000   -0.8000
   -0.4000    0.8000   -0.4000    0.2000
   -0.8000   -0.4000    0.2000    0.4000
R =
   -5.0000    2.0000   -1.0000
         0   -4.0000    1.0000
         0         0   -2.0000
         0         0         0

Best Answer

MATLAB's QR decomposition is computed using Householder transformations, which is generally more numerically advantageous.

Related Solutions

MATLAB: How to compare the elements of a matrix and fill another matrix with the elements of a vector

Since your t matrix is full of 0 and 1 you can take advantage of that and just do this

a = [0.9 0.8 0.7 0.6 0.5];
b = [0.1 0.2 0.3 0.4 0.5];
t = ff2n(n);
t1 = t.*a
t1 =
         0         0         0         0         0
         0         0         0         0    0.5000
         0         0         0    0.6000         0
         0         0         0    0.6000    0.5000
         0         0    0.7000         0         0
         0         0    0.7000         0    0.5000
         0         0    0.7000    0.6000         0
         0         0    0.7000    0.6000    0.5000
         0    0.8000         0         0         0
         0    0.8000         0         0    0.5000
         0    0.8000         0    0.6000         0
         0    0.8000         0    0.6000    0.5000
         0    0.8000    0.7000         0         0
         0    0.8000    0.7000         0    0.5000
         0    0.8000    0.7000    0.6000         0
         0    0.8000    0.7000    0.6000    0.5000
    0.9000         0         0         0         0
    0.9000         0         0         0    0.5000
    0.9000         0         0    0.6000         0
    0.9000         0         0    0.6000    0.5000
    0.9000         0    0.7000         0         0
    0.9000         0    0.7000         0    0.5000
    0.9000         0    0.7000    0.6000         0
    0.9000         0    0.7000    0.6000    0.5000
    0.9000    0.8000         0         0         0
    0.9000    0.8000         0         0    0.5000
    0.9000    0.8000         0    0.6000         0
    0.9000    0.8000         0    0.6000    0.5000
    0.9000    0.8000    0.7000         0         0
    0.9000    0.8000    0.7000         0    0.5000
    0.9000    0.8000    0.7000    0.6000         0
    0.9000    0.8000    0.7000    0.6000    0.5000
t2 = not(t.*a).*b
t2 =
    0.1000    0.2000    0.3000    0.4000    0.5000
    0.1000    0.2000    0.3000    0.4000         0
    0.1000    0.2000    0.3000         0    0.5000
    0.1000    0.2000    0.3000         0         0
    0.1000    0.2000         0    0.4000    0.5000
    0.1000    0.2000         0    0.4000         0
    0.1000    0.2000         0         0    0.5000
    0.1000    0.2000         0         0         0
    0.1000         0    0.3000    0.4000    0.5000
    0.1000         0    0.3000    0.4000         0
    0.1000         0    0.3000         0    0.5000
    0.1000         0    0.3000         0         0
    0.1000         0         0    0.4000    0.5000
    0.1000         0         0    0.4000         0
    0.1000         0         0         0    0.5000
    0.1000         0         0         0         0
         0    0.2000    0.3000    0.4000    0.5000
         0    0.2000    0.3000    0.4000         0
         0    0.2000    0.3000         0    0.5000
         0    0.2000    0.3000         0         0
         0    0.2000         0    0.4000    0.5000
         0    0.2000         0    0.4000         0
         0    0.2000         0         0    0.5000
         0    0.2000         0         0         0
         0         0    0.3000    0.4000    0.5000
         0         0    0.3000    0.4000         0
         0         0    0.3000         0    0.5000
         0         0    0.3000         0         0
         0         0         0    0.4000    0.5000
         0         0         0    0.4000         0
         0         0         0         0    0.5000
         0         0         0         0         0
t1+t2
ans =
    0.1000    0.2000    0.3000    0.4000    0.5000
    0.1000    0.2000    0.3000    0.4000    0.5000
    0.1000    0.2000    0.3000    0.6000    0.5000
    0.1000    0.2000    0.3000    0.6000    0.5000
    0.1000    0.2000    0.7000    0.4000    0.5000
    0.1000    0.2000    0.7000    0.4000    0.5000
    0.1000    0.2000    0.7000    0.6000    0.5000
    0.1000    0.2000    0.7000    0.6000    0.5000
    0.1000    0.8000    0.3000    0.4000    0.5000
    0.1000    0.8000    0.3000    0.4000    0.5000
    0.1000    0.8000    0.3000    0.6000    0.5000
    0.1000    0.8000    0.3000    0.6000    0.5000
    0.1000    0.8000    0.7000    0.4000    0.5000
    0.1000    0.8000    0.7000    0.4000    0.5000
    0.1000    0.8000    0.7000    0.6000    0.5000
    0.1000    0.8000    0.7000    0.6000    0.5000
    0.9000    0.2000    0.3000    0.4000    0.5000
    0.9000    0.2000    0.3000    0.4000    0.5000
    0.9000    0.2000    0.3000    0.6000    0.5000
    0.9000    0.2000    0.3000    0.6000    0.5000
    0.9000    0.2000    0.7000    0.4000    0.5000
    0.9000    0.2000    0.7000    0.4000    0.5000
    0.9000    0.2000    0.7000    0.6000    0.5000
    0.9000    0.2000    0.7000    0.6000    0.5000
    0.9000    0.8000    0.3000    0.4000    0.5000
    0.9000    0.8000    0.3000    0.4000    0.5000
    0.9000    0.8000    0.3000    0.6000    0.5000
    0.9000    0.8000    0.3000    0.6000    0.5000
    0.9000    0.8000    0.7000    0.4000    0.5000
    0.9000    0.8000    0.7000    0.4000    0.5000
    0.9000    0.8000    0.7000    0.6000    0.5000
    0.9000    0.8000    0.7000    0.6000    0.5000

MATLAB: How to display a polar coordinates image in Cartesian

For a simple example, if rho is the following 2D array:

 rho =
         0    0.2000    0.4000    0.6000    0.8000    1.0000
         0    0.2000    0.4000    0.6000    0.8000    1.0000
         0    0.2000    0.4000    0.6000    0.8000    1.0000
         0    0.2000    0.4000    0.6000    0.8000    1.0000
         0    0.2000    0.4000    0.6000    0.8000    1.0000
         0    0.2000    0.4000    0.6000    0.8000    1.0000
         0    0.2000    0.4000    0.6000    0.8000    1.0000

and theta is the following 2D array:

 theta =
     0     0     0     0     0     0
    15    15    15    15    15    15
    30    30    30    30    30    30
    45    45    45    45    45    45
    60    60    60    60    60    60
    75    75    75    75    75    75
    90    90    90    90    90    90

Here is the example code:

[rho, theta] = meshgrid(0:0.2:1.0,0:15:90);
[X Y] = pol2cart(theta*pi/180,rho);
S = surf(X,Y,ones(size(X))); 
Picture = imread('autumn.tif');
set(S,'FaceColor','Texturemap','CData',Picture);
view(2);

Type

doc cart2pol

 doc function_name

in the MATLAB command window for more information of the commands used in the example code.

For more information on the surface properties, please look at the following URL:

https://www.mathworks.com/help/releases/R2017a/matlab/ref/primitivesurface-properties.html

For a transition from Cartesian coordinates to polar or cylindrical, review the following function:

http://www.mathworks.com/access/helpdesk/help/techdoc/ref/cart2pol.shtml

Best Answer

Related Solutions

MATLAB: How to compare the elements of a matrix and fill another matrix with the elements of a vector

MATLAB: How to display a polar coordinates image in Cartesian

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