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+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\FloatXmpTitle}{Float}
+\newcommand{\FloatXmpNumber}{9.27}
+%
+% =====================================================================
+\begin{page}{FloatXmpPage}{9.27 Float}
+% =====================================================================
+\beginscroll
+%
+
+\Language{} provides two kinds of floating point numbers.
+The domain \spadtype{Float} (abbreviation \spadtype{FLOAT})
+implements a model of arbitrary
+%-% \HDindex{floating point!arbitrary precision}{FloatXmpPage}{9.27}{Float}
+precision floating point numbers.
+The domain \spadtype{DoubleFloat} (abbreviation \spadtype{DFLOAT})
+is intended to make available
+%-% \HDindex{floating point!hardware}{FloatXmpPage}{9.27}{Float}
+hardware floating point arithmetic in \Language{}.
+%-% \HDexptypeindex{DoubleFloat}{FloatXmpPage}{9.27}{Float}
+The actual model of floating point that \spadtype{DoubleFloat} provides is
+system-dependent.
+For example, on the IBM system 370 \Language{} uses IBM double
+precision which has fourteen hexadecimal digits of precision or roughly
+sixteen decimal digits.
+Arbitrary precision floats allow the user to specify the
+precision at which arithmetic operations are computed.
+Although this is an attractive facility, it comes at a cost.
+Arbitrary-precision floating-point arithmetic typically takes
+twenty to two hundred times more time than hardware floating point.
+
+For more information about \Language{}'s numeric and graphic
+facilities,  see
+\downlink{``\ugGraphTitle''}{ugGraphPage} in Section \ugGraphNumber\ignore{ugGraph},
+\downlink{``\ugProblemNumericTitle''}{ugProblemNumericPage} in Section \ugProblemNumericNumber\ignore{ugProblemNumeric}, and
+\downlink{`DoubleFloat'}{DoubleFloatXmpPage}\ignore{DoubleFloat}.
+
+\beginmenu
+    \menudownlink{{9.27.1. Introduction to Float}}{ugxFloatIntroPage}
+    \menudownlink{{9.27.2. Conversion Functions}}{ugxFloatConvertPage}
+    \menudownlink{{9.27.3. Output Functions}}{ugxFloatOutputPage}
+    \menudownlink{{9.27.4. An Example: Determinant of a Hilbert Matrix}}{ugxFloatHilbertPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFloatIntroTitle}{Introduction to Float}
+\newcommand{\ugxFloatIntroNumber}{9.27.1.}
+%
+% =====================================================================
+\begin{page}{ugxFloatIntroPage}{9.27.1. Introduction to Float}
+% =====================================================================
+\beginscroll
+
+Scientific notation is supported for input and output
+%-% \HDindex{floating point!input}{ugxFloatIntroPage}{9.27.1.}{Introduction to Float}
+of floating point numbers.
+%-% \HDindex{scientific notation}{ugxFloatIntroPage}{9.27.1.}{Introduction to Float}
+A floating point number is written as a string of digits containing a
+decimal point optionally followed by the letter ``{\tt E}'', and then
+the exponent.
+\xtc{
+We begin by doing some calculations using arbitrary precision floats.
+The default precision is twenty decimal digits.
+}{
+\spadpaste{1.234}
+}
+\xtc{
+A decimal base for the exponent is assumed, so the number
+\spad{1.234E2}  denotes
+\texht{$1.234 \cdot  10^2$}{\spad{1.234 * 10**2}}.
+}{
+\spadpaste{1.234E2}
+}
+\xtc{
+The normal arithmetic operations are available for floating point
+numbers.
+}{
+\spadpaste{sqrt(1.2 + 2.3 / 3.4 ** 4.5)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFloatConvertTitle}{Conversion Functions}
+\newcommand{\ugxFloatConvertNumber}{9.27.2.}
+%
+% =====================================================================
+\begin{page}{ugxFloatConvertPage}{9.27.2. Conversion Functions}
+% =====================================================================
+\beginscroll
+
+\labelSpace{3pc}
+\xtc{
+You can use conversion
+(\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert})
+to go back and forth between \spadtype{Integer}, \spadtype{Fraction Integer}
+and \spadtype{Float}, as appropriate.
+}{
+\spadpaste{i := 3 :: Float \bound{i}}
+}
+\xtc{
+}{
+\spadpaste{i :: Integer \free{i}}
+}
+\xtc{
+}{
+\spadpaste{i :: Fraction Integer \free{i}}
+}
+\xtc{
+Since you are explicitly asking for a conversion, you must take
+responsibility for any loss of exactness.
+}{
+\spadpaste{r := 3/7 :: Float \bound{r}}
+}
+\xtc{
+}{
+\spadpaste{r :: Fraction Integer \free{r}}
+}
+\xtc{
+This conversion cannot be performed: use
+\spadfunFrom{truncate}{Float} or \spadfunFrom{round}{Float} if that
+is what you intend.
+}{
+\spadpaste{r :: Integer \free{r}}
+}
+
+\xtc{
+The operations \spadfunFrom{truncate}{Float} and \spadfunFrom{round}{Float}
+truncate  \ldots
+}{
+\spadpaste{truncate 3.6}
+}
+\xtc{
+and round
+to the nearest integral \spadtype{Float} respectively.
+}{
+\spadpaste{round 3.6}
+}
+\xtc{
+}{
+\spadpaste{truncate(-3.6)}
+}
+\xtc{
+}{
+\spadpaste{round(-3.6)}
+}
+\xtc{
+The operation \spadfunFrom{fractionPart}{Float} computes the fractional part of
+\spad{x}, that is, \spad{x - truncate x}.
+}{
+\spadpaste{fractionPart 3.6}
+}
+\xtc{
+The operation \spadfunFrom{digits}{Float} allows the user to set the
+precision.
+It returns the previous value it was using.
+}{
+\spadpaste{digits 40 \bound{d40}}
+}
+\xtc{
+}{
+\spadpaste{sqrt 0.2}
+}
+\xtc{
+}{
+\spadpaste{pi()\$Float \free{d40}}
+}
+\xtc{
+The precision is only limited by the computer memory available.
+Calculations at 500 or more digits of precision are not difficult.
+}{
+\spadpaste{digits 500 \bound{d1000}}
+}
+\xtc{
+}{
+\spadpaste{pi()\$Float \free{d1000}}
+}
+\xtc{
+Reset \spadfunFrom{digits}{Float} to its default value.
+}{
+\spadpaste{digits 20}
+}
+Numbers of type \spadtype{Float} are represented as a record of two
+integers, namely, the mantissa and the exponent where the base of the
+exponent is binary.
+That is, the floating point number \spad{(m,e)} represents the number
+\texht{$m \cdot 2^e$}{\spad{m * 2**e}}.
+A consequence of using a binary base is that decimal numbers can not, in
+general, be represented exactly.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFloatOutputTitle}{Output Functions}
+\newcommand{\ugxFloatOutputNumber}{9.27.3.}
+%
+% =====================================================================
+\begin{page}{ugxFloatOutputPage}{9.27.3. Output Functions}
+% =====================================================================
+\beginscroll
+
+A number of operations exist for specifying how numbers of type \spadtype{Float} are to be
+%-% \HDindex{floating point!output}{ugxFloatOutputPage}{9.27.3.}{Output Functions}
+displayed.
+By default, spaces are inserted every ten digits in the
+output for readability.\footnote{Note that you cannot include spaces
+in the input form of a floating point number, though you can use
+underscores.}
+
+
+\xtc{
+Output spacing can be modified with the \spadfunFrom{outputSpacing}{Float}
+operation.
+This inserts no spaces and then displays the value of \spad{x}.
+}{
+\spadpaste{outputSpacing 0; x := sqrt 0.2 \bound{x}\bound{os0}}
+}
+\xtc{
+Issue this to have the spaces inserted every \spad{5} digits.
+}{
+\spadpaste{outputSpacing 5; x \bound{os5}\free{x}}
+}
+\xtc{
+By default, the system displays floats in either fixed format
+or scientific format, depending on the magnitude of the number.
+}{
+\spadpaste{y := x/10**10 \bound{y}\free{x os5}}
+}
+\xtc{
+A particular format may be requested with the operations
+\spadfunFrom{outputFloating}{Float} and \spadfunFrom{outputFixed}{Float}.
+}{
+\spadpaste{outputFloating(); x \bound{of} \free{os5 x}}
+}
+\xtc{
+}{
+\spadpaste{outputFixed(); y \bound{ox} \free{os5 y}}
+}
+\xtc{
+Additionally, you can ask for \spad{n} digits to be displayed after the
+decimal point.
+}{
+\spadpaste{outputFloating 2; y \bound{of2} \free{os5 y}}
+}
+\xtc{
+}{
+\spadpaste{outputFixed 2; x \bound{ox2} \free{os5 x}}
+}
+\xtc{
+This resets the output printing to the default behavior.
+}{
+\spadpaste{outputGeneral()}
+}
+%
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFloatHilbertTitle}{An Example: Determinant of a Hilbert Matrix}
+\newcommand{\ugxFloatHilbertNumber}{9.27.4.}
+%
+% =====================================================================
+\begin{page}{ugxFloatHilbertPage}{9.27.4. An Example: Determinant of a Hilbert Matrix}
+% =====================================================================
+\beginscroll
+
+Consider the problem of computing the determinant of a \spad{10} by \spad{10}
+%-% \HDindex{Hilbert matrix}{ugxFloatHilbertPage}{9.27.4.}{An Example: Determinant of a Hilbert Matrix}
+Hilbert matrix.
+The \eth{\spad{(i,j)}} entry of a Hilbert matrix is given by
+\spad{1/(i+j+1)}.
+
+\labelSpace{2pc}
+\xtc{
+First do the computation using rational numbers to obtain the
+exact result.
+}{
+\spadpaste{a: Matrix Fraction Integer := matrix [[1/(i+j+1) for j in 0..9] for i in 0..9] \bound{a}}
+}
+\xtc{
+This version of \spadfunFrom{determinant}{Matrix} uses Gaussian
+elimination.
+}{
+\spadpaste{d:= determinant a \free{a}\bound{d}}
+}
+\xtc{
+}{
+\spadpaste{d :: Float \free{d}}
+}
+\xtc{
+Now use hardware floats. Note that a semicolon (;) is used
+to prevent the display of the matrix.
+}{
+\spadpaste{b: Matrix DoubleFloat := matrix [[1/(i+j+1\$DoubleFloat) for j in 0..9] for i in 0..9]; \bound{b}}
+}
+\xtc{
+The result given by hardware floats is correct only to four significant
+digits of precision.
+In the jargon of numerical analysis, the Hilbert matrix is said to be
+``ill-conditioned.''
+%-% \HDindex{matrix!ill-conditioned}{ugxFloatHilbertPage}{9.27.4.}{An Example: Determinant of a Hilbert Matrix}
+}{
+\spadpaste{determinant b \free{b}}
+}
+\xtc{
+Now repeat the computation at a higher precision using \spadtype{Float}.
+}{
+\spadpaste{digits 40 \bound{d40}}
+}
+\xtc{
+}{
+\spadpaste{c: Matrix Float := matrix [[1/(i+j+1\$Float) for j in 0..9] for i in 0..9]; \free{d40} \bound{c}}
+}
+\xtc{
+}{
+\spadpaste{determinant c \free{c}}
+}
+\xtc{
+Reset \spadfunFrom{digits}{Float} to its default value.
+}{
+\spadpaste{digits 20}
+}
+\endscroll
+\autobuttons
+\end{page}
+%