HP (Hewlett-Packard) 50g User Manual

Page 11-50

Function LU takes as input a square matrix A, and returns a lower-triangular 
matrix L, an upper triangular matrix U, and a permutation matrix P, in stack 
levels 3, 2, and 1, respectively.  The results L, U, and P, satisfy the equation 
P

⋅A = L⋅U.    When you call the LU function, the calculator performs a Crout LU 

decomposition of A using partial pivoting.
For example, in RPN mode:  

[[-1,2,5][3,1,-2][7,6,5]]  LU

produces:

3:[[7 0 0][-1 2.86 0][3 –1.57 –1]
2: [[1 0.86 0.71][0 1 2][0 0 1]]
1: [[0 0 1][1 0 0][0 1 0]]

In ALG mode, the same exercise will be shown as follows:

     

A square matrix is said to be orthogonal if its columns represent unit vectors that 
are mutually orthogonal.  Thus, if we let matrix U = [v

1

2

 … v

n

] where the v

i

,

i = 1, 2, …, n, are column vectors, and if v

i•

j

 = 

δ

ij

, where 

δ

ij

 is the Kronecker’s 

delta function, then U will be an orthogonal matrix.  This conditions also imply 
that U

⋅ U

T

 = I.

The Singular Value Decomposition (SVD) of a rectangular matrix A

m

×n

 consists in 

determining the matrices U, S, and V, such that A

m

×n

 = U

m

×m

⋅S

m

×n

⋅V

T

n

×n

,

where U and V are orthogonal matrices, and S is a diagonal matrix.  The 
diagonal elements of S are called the singular values of A and are usually 
ordered so that s

≥ s

, for i = 1, 2, …, n-1.  The columns [u

j

] of U and [v

j

] of 

V are the corresponding singular vectors.

In RPN, function SVD (Singular Value Decomposition) takes as input a matrix 
A

n

×m

, and returns the matrices U

n

×n

, V

m

×m

, and a vector s in stack levels 3, 2, 

and 1, respectively.  The dimension of vector s is equal to the minimum of the 
values n and m.  The matrices U and V are as defined earlier for singular value