SSL II USER'S GUIDE - Lahey Computer Systems

Table 3.4 Standard and component routines
Problem Basic
functions
Types
of matrix
Standard routines Component routines
Systems of
linear equations
Real general matrix LAX
(A22-11-0101)
Complex general LCX
matrix
(A22-15-0101)
Real symmetric LSIX
matrix
(A22-21-0101)
Real positive-
LSX
definite symmetric
matrix
(A22-51-0101)
Real general band LBX1
matrix
(A52-11-0101)
⎡Real
tridiagonal⎤
⎣⎢ matrix ⎦⎥
⎡ LTX ⎤
⎣⎢ (A52 - 11- 0501) ⎦⎥
Real symmetric LSBIX
band matrix (A52-21-0101)
Real positive-
LSBX
definite symmetric
band matrix
(A52-31-0101)
⎡Positive
- definite ⎤
⎢symmetric
⎥
⎢
⎣tridiagonal
matrix⎥
⎦
LSTX
(A52-31-0501)
(a) (b) (c) (d)
ALU
(A22-11-0202)
CLU
(A22-15-0202)
SMDM
(A22-21-0202)
SLDL
(A22-51-0202)
BLU1
(A52-11-0202)
SBMDM
(A52-21-0202)
SBDL
(A52-31-0202)
A = LDL T (3.7)
where: L is a lower triangular matrix and D is a
diagonal matrix.
Matrices are decomposed as shown in Table 3.5.
Table 3.5 Decomposed matrices
Kinds of matrices Contents of decomposed matrices
General matrices PA = LU
L: Lower triangular matrix
U: Unit upper triangular matrix
P is a permutation matrix.
Positive-definite
symmetric
matrices
A = LDL T
L: Unit lower triangular matrix
D: Diagonal matrix
To minimize calculation, the
diagonal matrix is given as D -1
• Pivoting and scaling
Let us take a look at decomposition of the nonsingular
matrix given by (3.6).
A = ⎡00
. 10 . ⎤
⎢ ⎥
⎣20
. 00 . ⎦
(3.8)
In this state, LU decomposition is impossible. And also
in the case of (3.9)
A = ⎡00001
. 10 . ⎤
⎢ ⎥
⎣10
. 10 . ⎦
(3.9)
Decomposing by floating point arithmetic with the only
three working digits (in decimal) will cause unstable
LUX
(A22-11-0302)
CLUX
(A22-15-0302)
MDMX
(A22-21-0302)
LDLX
(A22-51-0302)
BLUX1
(A52-11-0302)
BMDMX
(A52-21-0302)
BDLX
(A52-31-0302)
LAXR
(A32-11-0401)
LCXR
(A22-15-0401)
LSIXR
(A22-21-0401)
LSXR
(A22-51-0401)
LBX1R
(A52-11-0401)
LSBXR
(A52-31-0401)
LINEAR ALGEBRA
LUIV
(A22-11-0602)
CLUIV
(A22-15-0602)
LDIV
(A22-51-0702)
solutions. These unfavorable conditions can frequently
occur when the condition of a matrix is not proper. This
can be avoided by pivoting, which selects the element
with the maximum absolute value for the pivot.
In the case of (3.9), problems can be avoided by
exchanging the first row with the second row.
In order to perform pivoting, the method used to select
the maximum absolute value must be unique. By
multiplying all of the elements of a row by a large enough
constant, any absolute value of non-zero element in the
row can be made larger than those of elements in the
other rows.
Therefore, it is just as important to equilibrate the rows
and columns as it is to validly determine a pivot element
of the maximum size in pivoting. SSL II uses partial
pivoting with row equilibration.
The row equilibration is performed by scaling so that
the maximum absolute value of each row of the matrix
becomes 1. However, actually the values of the elements
are not changed in scaling; the scaling factor is used
when selecting the pivot.
Since row exchanges are performed in pivoting, the
history data is stored as the transposition vector. The
matrix decomposition which accompanies this partial
pivoting can be expressed as;
PA = LU (3.10)
Where P is the permutation matrix which performs row
exchanges.
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