Singular Value Decomposiyion
TI-89 calculator, approximate mode
T means transpose   sqrt = square root symbol ˆ

:svd()
:ClrIO
:[[ 3,2,4][5,6,8][9,1,0]]   ->a     Any matrix
:aT->at                         transpose
:at*a->ata
:eigVl(ata) ->k2          eigenvalues
:eigVc(ata) ->k1         eigenvectors
:diag( sqrt(k2)) -> e    make diagonal matri
:Disp ³e²,e
:Pause
:ClrIO
:k1T-> ³v²,v                  transpose
:Pause
:ClrIO
a*v^(-1)*e^(-1) ->u
:Pause
:ClrIO
:Disp ³u*e*v²,u*e*v           test
:EndPrgm


========
HP49G+ calculator used in RPN mode in FIX 4 number style

a= U*e*V  

U is a 3x3 matrix in which each column is unitary and the columns are 
right angle vectors.

e is a diagonal matrix in descending order of size

V  is a 3x3 matrix in which each column is unitary and the columns are 
right angle vectors.

Take an arbitrary 3x3 matrix

      | 3. 2. 4. |
a = | 5. 6. 8. |      
     | 9. 1. 0. |

Compute SVD directly

a  SVD

       |  -0.3845   -0.1410   -0.9128    |
u =  |  -0.7820   -0.4755    0.4030    |
      |  -0.4906     0 .8683   0.0725   |


       |  -0.6950   -0.4364   -0.5714   |
v =  |   0.7137   -0.3226   -0.6217    |
      |  -0.0869    0.8399    -0.5357   |

ee = [  13,6385   7.0259    0.7931 ]

Make ee diagonal         ee 3  DIAG->  

       |  13,6385        0           0         |
 e = |     0             7.0259     0         |
       |     0                 0       0.7931   |  

 a= U*e*V  
=======
Compute step by step

                                           | 3 5 9 |
at = transpose a =            | 2 6 1 |          a  TRN
                                        | 4 8 0 |

                |  115   45  52   |
at*a  =     |    45   41  56   |
                |    52   56   80 |

ata=at*a 

ata  EGV    

Results

               |  1.0000        1.0000      -0.1035   |
s5 =        |   0.6279      -0.4520       1.0000   |
               |    0.8222      -0.8711     -0.6377   |


s6 = [  186.0077    49.3632    0.6291   ]

square root[186.0077]=13.6385
square root[49.3632  ]=  7.0259
square root[ 0.6291   ]=  0.7931

Members of diagonal

Create a progran named UNIT to do Gram-Schmidt process on matrix B to produce
three vectors in a matrix KK at right angles(Orthonormal).

<<  B TRN OBJ-> DROP {
 3. 1.  }   ->ARRY   ŒQ3¹ 
STO  { 3. 1. }  -> ARRY
ŒQ2¹  STO  { 3. 1.  }
->ARRY  ŒQ1¹  STO   Q1 Q1
ABS  /  ŒK1¹  STO Q2 Q2  
Q1  DOT Q1 Q1 DOT  / Q1  
*  -   ŒK2¹  STO  K2 K2 
ABS  /  ŒK2¹  STO  Q3 Q3
Q1 DOT Q1 Q1 DOT  /  Q1  
*  -  Q3  K2  DOT K2  K2
DOT /  K2 *  -  ŒK3¹  STO
K3 K3 ABS / ŒK3¹ STO
K1 K2 2. COL+ K3 3. 
COL+  ŒKK¹ STO KK >>

Results

         |   0.6950    0.7137    -0.0869   |
kk =  |   0.4364   -0.3226    0.8399    |
         |  0.5714    -0.6217  -0.5357    |

Transpose     kk TRN

v  =            |   0.6950       0.4364    0.5714    |
                  |   0.7137     -0.3226    -0.6217    |
                  |  -0.0869      0.8399    -0.5357    |


Compute u

a
v 1/x
e 1/x
----
* *

       |  -0.3845   -0.1410   -0.9128    |
u =  |  -0.7820   -0.4755    0.4030    |
      |  -0.4906     0 .8683   0.0725   |

=====================
TI-89 Gram-Schmidt Vector   does GS in TI language
==
There is much on www about applying SVD to math problems.

http://en.wikipedia.org/wiki/Singular_value_decomposition

http://www.uwlax.edu/faculty/will/svd/

http://planetmath.org/encyclopedia/SingularValueDecomposition.html

http://web.mit.edu/be.400/www/SVD/Singular_Value_Decomposition.htm

http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html

http://www.kwon3d.com/theory/jkinem/svd.html

http://mathworld.wolfram.com/SingularValueDecomposition.html

http://public.lanl.gov/mewall/kluwer2002.html

http://rkb.home.cern.ch/rkb/AN16pp/node265.html

http://ingrid.ldeo.columbia.edu/dochelp/StatTutorial/SVD/

http://www.cse.unr.edu/~bebis/CS791E/Notes/SVD.pdf

http://www.hpl.hp.com/techreports/97/HPL-97-96.pdf

http://cnx.org/content/m10739/latest/

http://www2.imm.dtu.dk/~pch/Projekter/tsvd.html

http://www.bmva.ac.uk/bmvc/1997/papers/081/node3.html#SECTION00030000000000000000

=====
Least squares using svd

3 equations, 2 unknowns

a =  | -11  2  |               b= | 0 |
       |   2  3   |                   | 7  |
       |  2  -1  |                   | 5  |

Apply svd to a

u  =   |  -0.9737    -0.0939 |
        |   0.1327     -0.9743 |
        |   0.1854      0.2046 |

v=    | 0.9885   -0.1511   |
       | -0.1511  -0.9885   |

ee= [ 11.4783   3.3599 ]  = [e1   e2 ]

d=  | 1/e1     0        |
     |    0       1/e2   |

T is transpose


xn= vT*d*uT *b  

xn= |  0.421 |    x1  x2   solution
      |  1.684  |