{\tt B300 RSRTNT:} {\bf An Integral of type} $R(x,\sqrt{a+bx+cx^2})$
A new subroutine subprogram {\tt B300 RSRTNT}
(An Integral of type $R(x,\sqrt{a+bx+cx^2})$
written in Fortran, which calculates, based on indefinite integration,
the integral
$$ \displaystyle \int_u^v \frac{x^k dx}{(\sqrt{a+bx+cx^2})^n}$$
for $|k| \le 3$, $n=1,3$, and real values of the parameters,
has been submitted to MATHLIB. In addition,
a double-precision version {\tt DSRTNT} is provided on IBM and similar
computers.
{\tt D501 LEAMAX:} {\bf Constrained Non-Linear Least Squares and
Maximum Likelihood Estimation}
A new subroutine subprogram package {\tt D501 LEAMAX}
(Constrained non-linear least squares and maximum likelihood
estimation), written in Fortran, has been submitted to MATHLIB.
\par
On computers other than CDC and Cray, only the double-precision
version will be available.
On CDC and Cray computers, only the single-precision
version will be available.
\par
The purpose of this package is to provide subprograms which calculate
an approximation to a minimum of an objective function $\varphi$, with
respect to $n$ unknown parameters
% $\mathbf{a}=(a_1,a_2,\ldots,a_n)^{\mathbf{T}} \in \Bbb R^n$, for
the following three problems:
\begin{enumerate}
\item The general non-linear least squares problem
$$\varphi_S(\mathbf{a}) = \frac{1}{2}
\sum_{i=1}^m\, [f_i(\mathbf{a})]^2,$$
\item The least squares data fitting problem
$$\varphi_F(\mathbf{a}) = \frac{1}{2} \sum_{i=1}^m\,
\left[\frac{y_i-f(x_i,\mathbf{a})}{\sigma_i}\right]^2,$$
\item The maximum likelihood estimation
$$ \varphi_M(\mathbf{a}) = -\sum_{i=1}^m\, \ln (f(x_i,\mathbf{a})),$$
\end{enumerate}
subject to bounds on the variables of the form
$\underline{a}_j \le a_j \le \overline{a}_j \qquad (j=1,2,\ldots,n)$.
\par This package replaces the following subroutine subprograms,
which thus become obsolete:
\begin{tabular}{>{\tt}l>{\tt}l>{\tt}lll}
D507 & MINSQ & MATHLIB & Minimization of Sum of Squares of Functions \\
D508 & LINSQ & MATHLIB & Linear Least-squares Fit \\
D510 & FUMILI & MATHLIB & Fitting Chisquare and Likelihood Function \\
E208 & LSQ & KERNLIB & Least Square Polynomial Fit
\end{tabular}
They will be left MATHLIB or KERNLIB for a few months and will then
be deleted.
\par
The new subprograms offer in all three cases the possibility of
handling minimization problems which are subject to bounds on the
variables. A further advantage is that the solution of linear least
squares problems is calculated by orthogonal transformations
(QR-decomposition). In particular for ill-conditioned problems,
this should be prefered to solving the normal equations, which had
been used in subprograms {\tt D508 LINSQ} and {\tt E208 LSQ}.
{\tt E210 NORBAS:} {\bf Polynomial Splines / Normalized B-Splines}
A new subprogram package {\tt E210} {\tt NORBAS} (Polynomial splines /
Normalized B-splines), written in Fortran, has been submitted to
{\tt MATHLIB}.
\par
On computers other than CDC and Cray, only the double-precision version
version will be available. On CDC and Cray computers, only the
single-precision version will be available.
\par
The purpose of this package is to provide subprograms for various
calculations with polynomial splines in {\bf one} and in {\bf two}
dimensions.
The polynomial splines are represented as linear combinations of
normalized B-splines.
\par
The package {\tt NORBAS} contains function and subroutine
subprograms for solving the following problems:
\begin{DL}{123}
\item[{\bf (K)}] Calculation of a set of spline-knots in an interval for
normalized B-splines of a given degree.
\item[{\bf (B)}] Calculation of the function value, the value of a
derivative or the value of the integral of a normalized B-spline
with fixed degree and index over a set of spline-knots.
\item[{\bf (P)}] Calculation of the function value, the value of a
derivative or the value of the integral of a polynomial spline function
in B-spline representation with given coefficients.
\item[{\bf (I)}] Calculation of the coefficients of a polynomial
interpolation spline in B-spline representation to a
user-supplied data set.
\item[{\bf (A)}] Calculation of the coefficients of a polynomial least
squares approximation spline in B-spline representation to a
user-supplied data set.
\item[{\bf (V)}] Calculation of the coefficients of a polynomial
variation diminishing spline approximation in B-spline representation
to a user-supplied function.
\item[{\bf (D)}] Calculation, from given coefficients of a polynomial
spline in B-spline representation, of the corresponding coefficients
of its derivative in B-spline representation.
\end{DL}
In particular, these subprograms are useful
for solving polynomial spline interpolation and approximation
problems to a given data set, or to a user-supplied function in
one and two dimensions. They allow the user to choose the degree
of splines and the set of knots. The subprograms supply values of a
spline function, values of its derivatives and values of its integral.
In addition to the above mentioned applications the subprograms {\bf (B)}
for computing normalized B-splines are also useful for solving various
further problems by spline functions , e.g. differential equations
in one and two dimensions and integral equations. However these special
problems are not supported by corresponding subprograms of the package.
\subsection{Transfers from KERNLIB to MATHLIB}
In continuing the reorganization of the Program Library, the
following subprograms have been transferred from KERNLIB to
MATHLIB. Most of these subprograms have been revised, mainly in
order to obtain higher accuracy in double-precison mode on
IBM and similar computers.
\begin{tabular}{>{\tt}l>{\tt}l@{\qquad}l}
B101 & ATG & Arc Tangent Function, \\
C205 & RZERO & Zero of a Function of One Real Variable, \\
C312 & BESJ0 & Bessel Functions $J$ and $Y$ of Orders Zero and One, \\
C313 & BESI0 & Modified Bessel Functions $I$ and $K$ of
Orders Zero and One \\
C336 & SININT & Sine and Cosine Integrals, \\
C337 & EXPINT & Exponential Integral (see Section X.X), \\
D103 & GAUSS & Adaptive Gaussian Integration, \\
D209 & RKSTP & First-order Differential Equations
(Runge-Kutta) (see Section X.X), \\
G100 & PROB & Upper Tail Probability of Chi-Squared
\end{tabular}
The possibility of redefining the
action to be taken when certain specific error conditions are
detected in the subprograms {\tt C205 RZERO}, {\tt C312 BESJ0},
{\tt C313 BESI0}, {\tt C336 SININT}, {\tt C337 EXPINT},
{\tt D103 GAUSS} and {\tt G100 PROB}
is transferred from subprogram {\tt N001 KERSET} to
subprogram {\tt N002 MTLSET}. Error condition {\tt D209.1} in
{\tt C209 RKSTP} (now {\tt C200 RKSTP}) has been suppressed.
\par
We apologize for the inconveniences which may arise due to this
change.