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/* NOTICE: Change of Copyright Status
*
* The author of this module, Carsten Grammes, has expressed in
* personal email that he has no more interest in this code, and
* doesn't claim any copyright. He has agreed to put this module
* into the public domain.
*
* Lars Hecking 15-02-1999
*/
/*
* Matrix algebra, part of
*
* Nonlinear least squares fit according to the
* Marquardt-Levenberg-algorithm
*
* added as Patch to Gnuplot (v3.2 and higher)
* by Carsten Grammes
* Experimental Physics, University of Saarbruecken, Germany
*
* Previous copyright of this module: Carsten Grammes, 1993
*
*/
#include
"matrix.h"
#include
"alloc.h"
#include
"fit.h"
#include
"util.h"
/*****************************************************************/
#define
Swap
(
a
,
b
) {double _temp = (a); (a) = (b); (b) = _temp;}
/* HBB 20010424: unused: */
/* #define WINZIG 1e-30 */
/*****************************************************************
internal prototypes
*****************************************************************/
static
GP_INLINE
int
fsign
(
double
x
);
/*****************************************************************
first straightforward vector and matrix allocation functions
*****************************************************************/
/* allocates a double vector with n elements */
double
*
vec
(
int
n
)
{
double
*
dp
;
if
(
n
<
1
)
return
NULL
;
dp
=
gp_alloc
(
n
*
sizeof
(
double
),
"vec"
);
return
dp
;
}
/* allocates a double matrix */
double
*
*
matr
(
int
rows
,
int
cols
)
{
int
i
;
double
*
*
m
;
if
(
rows
<
1
||
cols
<
1
)
return
NULL
;
m
=
gp_alloc
(
rows
*
sizeof
(
m
[
0
]),
"matrix row pointers"
);
m
[
0
]
=
gp_alloc
(
rows
*
cols
*
sizeof
(
m
[
0
][
0
]),
"matrix elements"
);
for
(
i
=
1
;
i
<
rows
;
i
++
)
m
[
i
]
=
m
[
i
-
1
]
+
cols
;
return
m
;
}
void
free_matr
(
double
*
*
m
)
{
if
(
m
!=
NULL
) {
free
(
m
[
0
]);
free
(
m
);
}
}
double
*
redim_vec
(
double
*
*
v
,
int
n
)
{
if
(
n
<
1
)
*
v
=
NULL
;
else
*
v
=
gp_realloc
(
*
v
,
n
*
sizeof
((
*
v
)[
0
]),
"vec"
);
return
*
v
;
}
/* HBB: TODO: is there a better value for 'epsilon'? how to specify
* 'inline'? is 'fsign' really not available elsewhere? use
* row-oriented version (p. 309) instead?
*/
static
GP_INLINE
int
fsign
(
double
x
)
{
return
(
x
>
0
?
1
: (
x
<
0
) ?
-1
:
0
);
}
/*****************************************************************
Solve least squares Problem C*x+d = r, |r| = min!, by Given rotations
(QR-decomposition). Direct implementation of the algorithm
presented in H.R.Schwarz: Numerische Mathematik, Equation 7.33
If 'd == NULL', d is not accessed: the routine just computes the QR
decomposition of C and exits.
*****************************************************************/
void
Givens
(
double
*
*
C
,
double
*
d
,
double
*
x
,
int
N
,
int
n
)
{
int
i
,
j
,
k
;
double
w
,
gamma
,
sigma
,
rho
,
temp
;
double
epsilon
=
DBL_EPSILON
;
/*
* First, construct QR decomposition of C, by 'rotating away'
* all elements of C below the diagonal. The rotations are
* stored in place as Givens coefficients rho.
* Vector d is also rotated in this same turn, if it exists
*/
for
(
j
=
0
;
j
<
n
;
j
++
) {
for
(
i
=
j
+
1
;
i
<
N
;
i
++
) {
if
(
C
[
i
][
j
]) {
if
(
fabs
(
C
[
j
][
j
])
<
epsilon
*
fabs
(
C
[
i
][
j
])) {
/* find the rotation parameters */
w
=
-
C
[
i
][
j
];
gamma
=
0
;
sigma
=
1
;
rho
=
1
;
}
else
{
w
=
fsign
(
C
[
j
][
j
])
*
sqrt
(
C
[
j
][
j
]
*
C
[
j
][
j
]
+
C
[
i
][
j
]
*
C
[
i
][
j
]);
if
(
w
==
0
)
Eex3
(
"w = 0 in Givens(); Cjj = %g, Cij = %g"
,
C
[
j
][
j
],
C
[
i
][
j
]);
gamma
=
C
[
j
][
j
] /
w
;
sigma
=
-
C
[
i
][
j
] /
w
;
rho
=
(
fabs
(
sigma
)
<
gamma
) ?
sigma
:
fsign
(
sigma
) /
gamma
;
}
C
[
j
][
j
]
=
w
;
C
[
i
][
j
]
=
rho
;
/* store rho in place, for later use */
for
(
k
=
j
+
1
;
k
<
n
;
k
++
) {
/* rotation on index pair (i,j) */
temp
=
gamma
*
C
[
j
][
k
]
-
sigma
*
C
[
i
][
k
];
C
[
i
][
k
]
=
sigma
*
C
[
j
][
k
]
+
gamma
*
C
[
i
][
k
];
C
[
j
][
k
]
=
temp
;
}
if
(
d
) {
/* if no d vector given, don't use it */
temp
=
gamma
*
d
[
j
]
-
sigma
*
d
[
i
];
/* rotate d */
d
[
i
]
=
sigma
*
d
[
j
]
+
gamma
*
d
[
i
];
d
[
j
]
=
temp
;
}
}
}
}
if
(!
d
)
/* stop here if no d was specified */
return
;
/* solve R*x+d = 0, by backsubstitution */
for
(
i
=
n
-
1
;
i
>=
0
;
i
--
) {
double
s
=
d
[
i
];
for
(
k
=
i
+
1
;
k
<
n
;
k
++
)
s
+=
C
[
i
][
k
]
*
x
[
k
];
if
(
C
[
i
][
i
]
==
0
)
Eex
(
"Singular matrix in Givens()"
);
x
[
i
]
=
-
s
/
C
[
i
][
i
];
}
}
/* Given a triangular Matrix R, compute (R^T * R)^(-1), by forward
* then back substitution
*
* R, I are n x n Matrices, I is for the result. Both must already be
* allocated.
*
* Will only calculate the lower triangle of I, as it is symmetric
*/
void
Invert_RtR
(
double
*
*
R
,
double
*
*
I
,
int
n
)
{
int
i
,
j
,
k
;
/* fill in the I matrix, and check R for regularity : */
for
(
i
=
0
;
i
<
n
;
i
++
) {
for
(
j
=
0
;
j
<
i
;
j
++
)
/* upper triangle isn't needed */
I
[
i
][
j
]
=
0
;
I
[
i
][
i
]
=
1
;
if
(!
R
[
i
][
i
])
Eex
(
"Singular matrix in Invert_RtR"
);
}
/* Forward substitution: Solve R^T * B = I, store B in place of I */
for
(
k
=
0
;
k
<
n
;
k
++
) {
for
(
i
=
k
;
i
<
n
;
i
++
) {
/* upper half needn't be computed */
double
s
=
I
[
i
][
k
];
for
(
j
=
k
;
j
<
i
;
j
++
)
/* for j<k, I[j][k] always stays zero! */
s
-=
R
[
j
][
i
]
*
I
[
j
][
k
];
I
[
i
][
k
]
=
s
/
R
[
i
][
i
];
}
}
/* Backward substitution: Solve R * A = B, store A in place of B */
for
(
k
=
0
;
k
<
n
;
k
++
) {
for
(
i
=
n
-
1
;
i
>=
k
;
i
--
) {
/* don't compute upper triangle of A */
double
s
=
I
[
i
][
k
];
for
(
j
=
i
+
1
;
j
<
n
;
j
++
)
s
-=
R
[
i
][
j
]
*
I
[
j
][
k
];
I
[
i
][
k
]
=
s
/
R
[
i
][
i
];
}
}
}
void
lu_decomp
(
double
*
*
a
,
int
n
,
int
*
indx
,
double
*
d
)
{
int
i
,
imax
=
-1
,
j
,
k
;
/* HBB: added initial value, to shut up gcc -Wall */
double
large
,
dummy
,
temp
,
*
*
ar
,
*
*
lim
,
*
limc
,
*
ac
,
*
dp
,
*
vscal
;
dp
=
vscal
=
vec
(
n
);
*
d
=
1.0
;
for
(
ar
=
a
,
lim
=
&
(
a
[
n
]);
ar
<
lim
;
ar
++
) {
large
=
0.0
;
for
(
ac
=
*
ar
,
limc
=
&
(
ac
[
n
]);
ac
<
limc
;)
if
((
temp
=
fabs
(
*
ac
++
))
>
large
)
large
=
temp
;
if
(
large
==
0.0
)
int_error
(
NO_CARET
,
"Singular matrix in LU-DECOMP"
);
*
dp
++
=
1
/
large
;
}
ar
=
a
;
for
(
j
=
0
;
j
<
n
;
j
++
,
ar
++
) {
for
(
i
=
0
;
i
<
j
;
i
++
) {
ac
=
&
(
a
[
i
][
j
]);
for
(
k
=
0
;
k
<
i
;
k
++
)
*
ac
-=
a
[
i
][
k
]
*
a
[
k
][
j
];
}
large
=
0.0
;
dp
=
&
(
vscal
[
j
]);
for
(
i
=
j
;
i
<
n
;
i
++
) {
ac
=
&
(
a
[
i
][
j
]);
for
(
k
=
0
;
k
<
j
;
k
++
)
*
ac
-=
a
[
i
][
k
]
*
a
[
k
][
j
];
if
((
dummy
=
*
dp
++
*
fabs
(
*
ac
)) >=
large
) {
large
=
dummy
;
imax
=
i
;
}
}
if
(
j
!=
imax
) {
ac
=
a
[
imax
];
dp
=
*
ar
;
for
(
k
=
0
;
k
<
n
;
k
++
,
ac
++
,
dp
++
)
Swap
(
*
ac
,
*
dp
);
*
d
=
-
(
*
d
);
vscal
[
imax
]
=
vscal
[
j
];
}
indx
[
j
]
=
imax
;
if
(
*
(
dp
=
&
(
*
ar
)[
j
])
==
0
)
*
dp
=
1e-30
;
if
(
j
!=
n
-
1
) {
dummy
=
1
/ (
*
ar
)[
j
];
for
(
i
=
j
+
1
;
i
<
n
;
i
++
)
a
[
i
][
j
] *=
dummy
;
}
}
free
(
vscal
);
}
void
lu_backsubst
(
double
*
*
a
,
int
n
,
int
*
indx
,
double
*
b
)
{
int
i
,
memi
=
-1
,
ip
,
j
;
double
sum
,
*
bp
,
*
bip
,
*
*
ar
,
*
ac
;
ar
=
a
;
for
(
i
=
0
;
i
<
n
;
i
++
,
ar
++
) {
ip
=
indx
[
i
];
sum
=
b
[
ip
];
b
[
ip
]
=
b
[
i
];
if
(
memi
>=
0
) {
ac
=
&
((
*
ar
)[
memi
]);
bp
=
&
(
b
[
memi
]);
for
(
j
=
memi
;
j
<=
i
-
1
;
j
++
)
sum
-=
*
ac
++
*
*
bp
++
;
}
else
if
(
sum
)
memi
=
i
;
b
[
i
]
=
sum
;
}
ar
--
;
for
(
i
=
n
-
1
;
i
>=
0
;
i
--
) {
ac
=
&
(
*
ar
)[
i
+
1
];
bp
=
&
(
b
[
i
+
1
]);
bip
=
&
(
b
[
i
]);
for
(
j
=
i
+
1
;
j
<
n
;
j
++
)
*
bip
-=
*
ac
++
*
*
bp
++
;
*
bip
/= (
*
ar
--
)[
i
];
}
}
/*****************************************************************
Sum up squared components of a vector
In order to reduce rounding errors in summing up the entries
of a vector, we employ the Neumaier variant of the Kahan and
Babuska algorithm:
A. Neumaier, Rundungsfehleranalyse einiger Verfahren zur
Summation endlicher Summen, Z. angew. Math. Mechanik, 54:39-51, 1974
*****************************************************************/
double
sumsq_vec
(
int
n
,
const
double
*
x
)
{
int
i
;
double
s
;
double
c
=
0.0
;
if
((
x
==
NULL
)
||
(
n
==
0
))
return
0
;
s
=
x
[
0
]
*
x
[
0
];
for
(
i
=
1
;
i
<
n
;
i
++
) {
double
xi
=
x
[
i
]
*
x
[
i
];
double
t
=
s
+
xi
;
if
(
fabs
(
s
) >=
fabs
(
xi
))
c
+=
((
s
-
t
)
+
xi
);
else
c
+=
((
xi
-
t
)
+
s
);
s
=
t
;
};
s
+=
c
;
return
s
;
}
/*****************************************************************
Euclidean norm of a vector
*****************************************************************/
double
enorm_vec
(
int
n
,
const
double
*
x
)
{
return
sqrt
(
sumsq_vec
(
n
,
x
));
}
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