... a[i][j-1]=a[i][j]-dum*a[k][j];
a[i][mm]=0.0;
}
}
}
52
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0 -52 1-43108 -5)
Copyright (C) 1988-1992 ... += a[i][j]*x[j+k];
}
}
54
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFI...
... trade@cup.cam.ac.uk (outside North America).
Chapter 2. Solution of Linear
Algebraic Equations
2.0 Introduction
A set of linear algebraic equations looks like this:
a
11
x
1
+ a
12
x
2
+ a
13
x
3
+ ... equations as unknowns, and there is a good
chance of solving for a unique solution set of x
j
’s. Analytically, there can fail to
be a unique solution if one or more...
... 46
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0 -52 1-43108 -5)
Copyright (C) 1988-1992 by Cambridge ... (i.e., selection of a salubrious pivot element
for the division in equation 2.3.13) is absolutely essential for the stability of Crout’s
50
Chapter 2. Solution of Linear Algebrai...
... North America).
A
⋅
x = b
SVD solution
of A
⋅
x = c
solutions of
A
⋅
x = c′
solutions of
A
⋅
x = d
null
space
of A
SVD solution of
A
⋅
x = d
range of A
d
c
(b)
(a)
A
x
b
c′
Figure ... Hopkins
University Press),
§
5. 2.6. [5]
60
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING...
... 90
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0 -52 1-43108 -5)
Copyright (C) 1988-1992 by Cambridge ... case of a tridiagonal matrix was treated specially, because that
particular type of linear system admits a solution in only of order N operations,
rather than of order N
3
for the...
... 92
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0 -52 1-43108 -5)
Copyright (C) 1988-1992 by Cambridge ... double precision.
94
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0 -52 1-43108 -5)
Copyright...
... equations
(2.3.12)–(2.3.13),
L
ii
=
a
ii
−
i−1
k=1
L
2
ik
1/2
(2.9.4)
and
L
ji
=
1
L
ii
a
ij
−
i−1
k=1
L
ik
L
jk
j = i +1,i+2, ,N (2.9 .5)
92
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0 -52 1-43108 -5)
Copyright (C) 1988-1992 ... 96
Chapter 2. Solution of Linear Algebraic...
... 36
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0 -52 1-43108 -5)
Copyright (C) 1988-1992 by Cambridge ... case of N × N matrices, with M sets of right-hand
side vectors, in completely analogous fashion. The routine implemented below
is, of course, general.
38
Chapter 2. Solution of Lin...
... solve linear systems. In many applications only the
part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv.
98
Chapter 2. Solution of Linear Algebraic Equations
Sample ... not used for typical systems of linear equations. However, we will
meet special cases where QR is the method of choice.
100
Chapter 2. Solution of Linear Alg...
... 42
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0 -52 1-43108 -5)
Copyright (C) 1988-1992 by Cambridge ... is called backsubstitution.Thecom-
bination of Gaussian elimination and backsubstitution yields a solution to the set
of equations.
The advantage of Gaussian elimination and backsub...