Chapter 8 Machine representation
This chapter is based on a lecture given by Professor Sivan Leviyang on January 28, 2016 for MATH-504 Numerical Methods at Georgetown University.
8.1 Binary numbers
Following Sauer (2011), binary numbers are expressed as
\[ \ldots b_{2}b_{1}b_{0}.b_{-1}b_{-2}\ldots, \]
where each binary digit, or bit, is 0 or 1. The decimal (base 10) equivalent to the number is
\[ \ldots b_{2}2^{2}+b_{1}2^{1}+b_{0}2^{0}+b_{-1}2^{-1}+b_{-2}2^{-2}\ldots. \]
A binary number with no fractional part may be converted to the corresponding decimal integer by adding (the nonnegative) powers of 2. If the \(j\text{th}\) bit is 1, then the sum should include the term \(2^{j}\).
Example 8.1 The binary number \(10001\) has \(1\text{s}\) in positions 0 and 4. Denoting by \(\left(\cdot\right)_{b}\) a number in base \(b\), we have
\[ \left(10001\right)_{2}=\sum_{j=0}^{4}b_{j}2^{j} =2^{0}+0+0+0+2^{4} =17. \]If the fractional part of a binary number is finite (a terminating base 2 expansion), we can proceed in the same fashion.
Example 8.2 The binary number \(0.101\) has \(1\text{s}\) in positions \(-1\) and \(-3\), so that
\[ \left(0.101\right)_{2}=2^{-1}+2^{-3} =\frac{1}{2}+\frac{1}{8} =\frac{5}{8}. \]The situation is more involved if the fractional part does not terminate, but that discussion is beyond the scope of this chapter.
8.2 Integers
Suppose that we encode an integer using 32 bits (equivalently, 4 bytes), each of which may be 0 or 1. We allocate a single bit \(s\) to hold the sign, and we denote the \(k\text{th}\) bit by \(i_{k}\). Then, an integer \(x\) can be written as
\[ x=\left(-1\right)^{s}\cdot\left(i_{0}2^{0}+i_{1}2^{1}+i_{2}2^{2}+\cdots+i_{30}2^{30}\right) =\left(-1\right)^{s}\sum_{k=0}^{30}i_{k}2^{k}, \]
and we can store this representation as the vector
\[ \begin{bmatrix} s & i_{0} & i_{1} & \cdots & i_{29} & i_{30} \end{bmatrix}, \]
where \(i_{k}=1\) if the \(k\text{th}\) term is included in the sum and 0 if it is not. R uses signed 32-bit integers, so that the maximum integer that can be represented is \(2^{31}-1=2147483647\), which we can confirm by inspecting the .Machine variable.
.Machine$integer.max## [1] 2147483647Observe that this is equivalent to the maximum number representable under the scheme described above (though R does not use this exact scheme).
sum(2 ^ seq(0, 30))## [1] 2147483647Now, this encoding is inefficient because it represents zero in two ways (zero is unsigned, so we may have \(s=0\) or \(s=1\)), hence we are losing a bit. Suppose instead that we allocate all 32 bits to terms in the above summation, and by convention subtract \(2^{31}\) from the sum, so that negative integers can be represented. Then, under this scheme, \[ x=\left(i_{0}2^{0}+i_{1}2^{1}+i_{2}2^{2}+\cdots+i_{30}2^{30}+i_{31}2^{31}\right)-2^{31} =\sum_{k=1}^{32}i_{k}2^{k}-2^{31}. \]
To represent 1, we will have \(i_{0}=1\), \(i_{31}=1\), and \(i_{k}=0\) for \(k\in\left\{ 1,2,\ldots,30\right\}\). The next consecutive integer is 2, which we represent with \(i_{1}=1\), \(i_{31}=1\), and \(i_{k}=0\) for \(k\in\left\{ 0\right\} \cup\left\{ 2,3,\ldots,30\right\}\). We represent 3 as \(i_{0}=1\), \(i_{1}=1\), \(i_{31}=1\), and \(i_{k}=0\) for \(k\in\left\{ 2,3,\ldots,30\right\}\). We proceed in this fashion until \(i_{k}=1\) for \(k\in\left\{ 0,1,\ldots,31\right\}\), i.e.,
\[ \begin{align*} x_{\max} & = \left(2^{0}+2^{1}+2^{2}+\cdots+2^{30}+2^{31}\right)-2^{31} \\ & = 2^{0}+2^{1}+2^{2}+\cdots+2^{30} \\ & = 2147483647 \\ & = 2^{31}-1, \end{align*} \]
which is the largest integer that can be stored under this encoding scheme. The largest negative integer that can be stored occurs when each \(i_{k}\) is 0, and this is \(-2^{31}\). Zero is represented by setting \(i_{31}=1\) and \(i_{k}=0\) for \(k\in\left\{ 0,1,\ldots,30\right\}\). Thus, we see that there are a variety of possible encoding schemes for integers, each with its own considerations.
8.3 Floating-point numbers
The integer encoding schemes discussed in the previous section have two immediate drawbacks: they can only represent integers, and in general a \(k\)-bit encoding scheme cannot represent numbers larger than \(2^k\) (provided that all intermediate numbers must be representable). We now introduce floating-point numbers, which address these issues. Under the IEEE 754 Floating Point Standard, numbers are encoded as 64-bit (8-byte) words of the form
\[ \begin{bmatrix} s & e_{1} & e_{2} & \cdots & e_{11} & b_{1} & b_{2} & \cdots & b_{52} \end{bmatrix}, \]
where \(s\) is the sign bit, the 11 \(e_{k}\) bits represent the exponent, and the 52 \(b_{j}\) bits represent the mantissa. Under this scheme, the representation of \(x\in\mathbb{R}\) is
\[ x=\left(-1\right)^{s}\cdot1.\boxed{b_{1}b_{2}\ldots b_{52}}\cdot2^{\boxed{e_{1}e_{2}\ldots e_{11}}-1023}, \]
where \(\boxed{b_{1}b_{2}\dots b_{52}}\) and \(\boxed{e_{1}e_{2}\ldots e_{11}}\) are the concatenations of the \(b_{j}\) bits and \(e_{k}\) bits, respectively.
We first consider the exponent. More precisely, the \(e_{k}\) bits represent the positive binary integer that is the sum of the exponent and the bias \(2^{10}-1=1023\) (for exponents between -1022 and 1023). Letting \(p=\left(\boxed{e_{1}e_{2}\ldots e_{11}}\right)_{2}-\left(1023\right)_{10}\) be the decimal exponent, we will consider some examples.
| \(p\) | \(p+1023\) | \(\left(p+1023\right)_{2}\) | \(\boxed{e_{1}e_{2}\ldots e_{11}}\) |
|---|---|---|---|
| \(-1\) | \(1022\) | \(11\,1111\,1110\) | \(\mathtt{011\,1111\,1110}\) |
| \(0\) | \(1023\) | \(11\,1111\,1111\) | \(\mathtt{011\,1111\,1111}\) |
| \(1\) | \(1024\) | \(100\,0000\,0000\) | \(\mathtt{100\,0000\,0000}\) |
| \(2\) | \(1025\) | \(100\,0000\,0001\) | \(\mathtt{100\,0000\,0001}\) |
| \(17\) | \(1040\) | \(100\,0001\,0000\) | \(\mathtt{100\,0001\,0000}\) |
Thus, given some \(\boxed{e_{1}e_{2}\ldots e_{11}}\), we can recover the exponent \(p\) by subtracting the bias.
We now consider the mantissa. Sauer (2011) describes the normalized floating-point representation of a binary number as being “left-justified, meaning that the leftmost 1 is shifted just to the left of the radix point” (generalization of a decimal point). We can change the exponent to compensate for shifting the radix, i.e., the radix “floats.” Observe that this is the same process used to convert a decimal number to scientific notation, e.g., \(256=2.56\cdot 10^{2}\).
Now, the \(1\) in \(1.\boxed{b_{1}b_{2}\ldots b_{52}}\) is assumed, not stored. More formally, we have
\[ x=\left(-1\right)^{s}\left(1+\sum_{j=1}^{52}b_{j}2^{-j}\right)\cdot2^{p-1023}. \]
Thus, we see that the term \(2^{-j}\) is included in the representation of \(x\) precisely when \(b_{j}=1\) (the term is \(2^{-j}\) and not \(2^{j}\) because the \(b_{j}\text{th}\) bit occurs to the right of the radix).
Example 8.3 The decimal number 9 is equivalent to \(\left(1001\right)_{2}\). We can represent this as a floating-point number by shifting the radix point by 3 bits and setting the exponent to 3, i.e.,
\[ \left(9\right)_{10}=\left(1001\right)_{2}=1.001\cdot2^3, \]
so that \(s=0\), \(b_{j}=1\) for \(j=3\), and \(p=3\implies p+1023=1026\implies e_{k}=1\) for \(k\in\left\{1,10\right\}\). As a check, note that
\[ \left(-1\right)^{s}\left(1+\sum_{j=1}^{52}b_{j}2^{-j}\right)\cdot 2^{p-1023} = \left(1+2^{-3}\right)\cdot 2^{1026-1023} = 1.125\cdot 2^{3} = 9. \]Example 8.4 The decimal number 17.625 is equivalent to \(\left(10001.101\right)_{2}\). We can represent this as a floating-point number by shifting the radix point by 4 bits, i.e.,
\[ \left(17.625\right)_{10}=\left(10001.101\right)_{2}=1.0001101\cdot2^4, \]
so that \(s=0\), \(b_{j}=1\) for \(j\in\left\{4,5,7\right\}\), and \(p=4\implies p+1023=1027\implies e_{k}=1\) for \(k\in\left\{1,10,11\right\}\). As a check, note that
\[ \left(-1\right)^{s}\left(1+\sum_{j=1}^{52}b_{j}2^{-j}\right)\cdot 2^{p-1023} = \left(1+2^{-4}+2^{-5}+2^{-7}\right)\cdot 2^{1027-1023} = 1.1015625\cdot 2^{4} = 17.625. \]8.3.1 Special exponent values
Now, the exponent value \(\left(2047\right)_{10}=\left(111\,1111\,1111\right)_{2}\) is reserved to represent infinity if every \(b_{j}\) is zero, i.e., if the mantissa bits are all zero, and to represent NaN (Not a Number) otherwise. The exponent value 0 is used to represent subnormal floating point numbers, or those numbers where the left-most bit is not assumed to be 1. We summarize a variety of special cases below.
| \(s\) | \(\boxed{e_{1}e_{2}\ldots e_{11}}\) | \(\boxed{b_{1}b_{2}\ldots b_{52}}\) | value |
|---|---|---|---|
| \(\mathtt{0}\) | \(\mathtt{111\,1111\,1111}\) | \(b_{j}=0\,\forall j\) | \(+\infty\) |
| \(\mathtt{1}\) | \(\mathtt{111\,1111\,1111}\) | \(b_{j}=0\,\forall j\) | \(-\infty\) |
| \(\mathtt{0}\) | \(\mathtt{111\,1111\,1111}\) | \(b_{52}=1\) | NaN |
| \(\mathtt{0}\) | \(\mathtt{000\,0000\,0000}\) | \(\exists\,j:b_{j}=1\) | subnormal |
| \(\mathtt{0}\) | \(\mathtt{000\,0000\,0000}\) | \(b_{j}=0\,\forall j\) | \(+0\) |
| \(\mathtt{1}\) | \(\mathtt{000\,0000\,0000}\) | \(b_{j}=0\,\forall j\) | \(-0\) |
Thus, the smallest non-reserved value the exponent bits can take is \(\mathtt{000\,0000\,0001}\), which corresponds to an exponent of \(2^{0}-1023=-1022\). The largest non-reserved value the exponent bits can take is \(\mathtt{111\,1111\,1110}\), which corresponds to an exponent of \(\sum_{k=1}^{10}2^{k}-1023=1023\).
It follows that the range of the exponent is \(\left(-2^{10}+2,2^{10}-1\right)\), so that the largest number that can be represented using the double precision floating-point encoding is roughly \(2^{2^{10}-1}=2^{1023}\). To be precise, we must also consider the mantissa. If all the mantissa bits are 1, then
\[ \sum_{j=1}^{52}b_{j}2^{-j}=\sum_{j=1}^{52}2^{-j}=\frac{1}{2^{1}}+\frac{1}{2^{2}}+\cdots+\frac{1}{2^{52}}, \]
which is a geometric series of the form \(a+ar+ar^{2}+ar^{3}+\cdots\), with \(a=r=1/2\). The sum is finite, but we can view it as the sum of the first \(n\) terms of an infinite sum, which has the well-known formula
\[ \sum_{k=0}^{n-1}ar^{k}=a\left(\frac{1-r^{n}}{1-r}\right). \]
Thus,
\[ \sum_{j=1}^{52}2^{-j} = \frac{1}{2}\left(\frac{1-\left(1/2\right)^{52}}{1-1/2}\right) = 1-\left(\frac{1}{2}\right)^{52} = 1-2^{-52}, \]
so that
\[ x_{\max}=\left(-1\right)^{s}\left(1+\sum_{j=1}^{52}b_{j}2^{-j}\right)\cdot 2^{p-1023} = \left(1+1-2^{-52}\right)\cdot 2^{1023} = \left(2-2^{-52}\right)\cdot 2^{1023}. \]
We can compute this quantity.
(2 - 2 ^ -52) * 2 ^ 1023## [1] 1.797693e+308We see that this is equal to the maximum value of a double-precision floating point number.
.Machine$double.xmax## [1] 1.797693e+3088.3.2 Limitations
Floating-point representation has the advantage of being able to represent a large range of numbers, including on very different scales. It also has limitations. For example, it is not possible to represent every real number exactly. For example, \(\sqrt{2}\) does not have a finite decimal expansion, and extremely large or small numbers cannot be represented exactly due to the defined (and finite) number of bits available for representation.
For example, it can be shown that the largest consecutive integer that can be stored exactly is \(2^{53}\). If we add 1 to this number, R will be unable to differentiate between them.
2 ^ 53 == (2 ^ 53 + 1)## [1] TRUEFloating-point arithmetic is also subject to round-off error, which is especially problematic when adding two numbers on very different scales.
8.3.3 Floating-point error
We now consider the accuracy of floating-point representation. We have 52 stored mantissa bits plus 1 implicit leading bit to store the precision of \(x\), which is equivalent to 15-17 decimal bits. The relative roundoff error in representing \(x\) is \(\left|\text{fl}\left(x\right)-x\right|/\left|x\right|\).
Example 8.5 Suppose that \(x=12345678901234567890\). Assuming that 16 decimal bits of precision are available to represent \(x\), we have \(\text{fl}\left(x\right)=12345678901234560000\), so that the relative roundoff error is
\[ \frac{\left|\text{fl}\left(x\right)-x\right|}{\left|x\right|}=\frac{\left|7890\right|}{\left|x\right|}\approx\frac{10^{4}}{10^{20}}=10^{-16}. \]Alternatively, \(\epsilon_{\text{mach}}\) is the smallest positive number for which \(\text{fl}\left(1+\epsilon_{\text{mach}}\right)\neq1\). Under the IEEE double precision floating-point standard, machine epsilon is \(2^{-52}\approx10^{-16}\).
2 ^ -52 == .Machine$double.eps## [1] TRUEIn practice, R cannot distinguish between some \(x\in\mathbb{R}\) and \(x+c\) for some \(c<\epsilon_{\text{mach}}\).
1 == 1 + 2 ^ -53## [1] TRUEThe relative roundoff error in representing some \(x\neq0\) will be at most \(\epsilon_{\text{mach}}\), i.e.,
\[ \frac{\left|\text{fl}\left(x\right)-x\right|}{\left|x\right|}\leq\epsilon_{\text{mach}}. \]
In the worst case, i.e., equality, we will have \(\left|\text{fl}\left(x\right)-x\right|=\left|x\right|\epsilon_{\text{mach}}\). If \(\text{fl}\left(x\right)\geq x\), then \(\left|\text{fl}\left(x\right)-x\right|\geq0\), so that this expression becomes
\[ \left|\text{fl}\left(x\right)-x\right|=\left|x\right|\epsilon_{\text{mach}} \implies\text{fl}\left(x\right)-x=\left|x\right|\epsilon_{\text{mach}} \implies\text{fl}\left(x\right)=x+\left|x\right|\epsilon_{\text{mach}}. \]
If \(\text{fl}\left(x\right)<x\), then \(\left|\text{fl}\left(x\right)-x\right|<0\), so that the expression becomes
\[ \left|\text{fl}\left(x\right)-x\right|=\left|x\right|\epsilon_{\text{mach}} \implies-\left(\text{fl}\left(x\right)-x\right)=\left|x\right|\epsilon_{\text{mach}} \implies\text{fl}\left(x\right)=x-\left|x\right|\epsilon_{\text{mach}}, \]
which we can express compactly as \(\text{fl}\left(x\right)=x\pm\left|x\right|\epsilon_{\text{mach}}\). If \(x\geq0\), then the right side of this expression becomes
\[ x\pm x\epsilon_{\text{mach}}=x\left(1\pm\epsilon_{\text{mach}}\right). \]
If \(x<0\), the right side of the expression becomes
\[ x\pm\left(-x\right)\epsilon_{\text{mach}}=x\pm x\epsilon_{\text{mach}} =x\left(1\pm\epsilon_{\text{mach}}\right), \]
hence \(\text{fl}\left(x\right)=x\left(1\pm\epsilon_{\text{mach}}\right)\) for all real \(x\). Thus, the relative error of floating-point representation is bounded by \(x\left(1\pm\epsilon_{\text{mach}}\right)\).
Example 8.6 Suppose \(x,y\in\mathbb{R}\), and consider the floating-point representation of \(\left(x+y\right)^{2}\). We have
\[\begin{align*} \text{fl}\left(\left(x+y\right)^{2}\right) & =\text{fl}\left(\left(\text{fl}\left(x\right)+\text{fl}\left(y\right)\right)^{2}\right) \\ & =\text{fl}\left(\left(\text{fl}\left(x\right)+\text{fl}\left(y\right)\right)\cdot\left(\text{fl}\left(x\right)+\text{fl}\left(y\right)\right)\right) \\ & =\text{fl}\left(\left(x\left(1+\epsilon_{1}\right)+y\left(1+\epsilon_{2}\right)\right)\cdot\left(x\left(1+\epsilon_{1}\right)+y\left(1+\epsilon_{2}\right)\right)\right) \\ & =\text{fl}\left(x^{2}\left(1+\epsilon_{1}\right)^{2}\left(1+\epsilon_{3}\right)+y^{2}\left(1+\epsilon_{2}\right)^{2}\left(1+\epsilon_{4}\right)+2xy\left(1+\epsilon_{1}\right)\left(1+\epsilon_{2}\right)\left(1+\epsilon_{5}\right)\right) \\ & =\left[x^{2}\left(1+\epsilon_{1}\right)^{2}\left(1+\epsilon_{3}\right)+y^{2}\left(1+\epsilon_{2}\right)^{2}\left(1+\epsilon_{4}\right)+2xy\left(1+\epsilon_{1}\right)\left(1+\epsilon_{2}\right)\left(1+\epsilon_{5}\right)\right] \\ &\quad\times\left(1+\epsilon_{6}\right), \end{align*}\]where the \(\epsilon_{i}\) terms are specific to the respective floating-point representations of each quantity and are on the order of \(\epsilon_{\text{mach}}\). This expression can be simplified as
\[ \text{fl}\left(\left(x+y\right)^{2}\right)=\left(x+y\right)^{2}\left(1+c\epsilon_{\text{mach}}\right), \]
where \(c\approx4\) (and in particular, \(c>1\)).We see from this example that floating-point errors can add up.
Example 8.7 The Vandermonde matrix has the form
\[ \mathbf{A}=\begin{bmatrix} 1 & \alpha_{1} & \alpha_{1}^{2} & \alpha_{1}^{3} & \cdots & \alpha_{1}^{n-1} \\ 1 & \alpha_{2} & \alpha_{2}^{2} & \alpha_{2}^{3} & \cdots & \alpha_{2}^{n-1} \\ 1 & \alpha_{3} & \alpha_{3}^{2} & \alpha_{3}^{3} & \cdots & \alpha_{3}^{n-1} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \alpha_{m} & \alpha_{m}^{2} & \alpha_{m}^{3} & \cdots & \alpha_{m}^{n-1} \\ \end{bmatrix}. \]
Let \(\mathbf{b}\in\mathbb{R}^{n}\), and consider solving \(\mathbf{A}\mathbf{x}=\mathbf{b}\). When we invert \(\mathbf{A}\) to find the solution \(\mathbf{x}=\mathbf{A}^{-1}\mathbf{b}\), we will mix quantities on very different scales, and floating-point error will propagate through our calculation.
Suppose that \(m=n=20\) and
\[ \boldsymbol{\alpha}=\left(\frac{1}{20},\frac{2}{20},\cdots,\frac{19}{20},1\right). \]
Letting \(\mathbf{x}=\left(1,2,\ldots,20\right)\), we will compute \(\mathbf{b}\).options(digits = 16)
n <- 20
x <- seq(n)
alpha <- x / n
A <- t(vapply(alpha, `^`, numeric(n), seq(0, n - 1)))
b <- A %*% xNow we pretend that we do not know \(\mathbf{x}\) and will solve for it. Comparing \(\mathbf{x}\) to the numerical solution \(\hat{\mathbf{x}}\), we see that some elements are close, but in most cases the solution is highly inaccurate. We will see in the future that this inaccuracy is intimately connected to the condition number \(\kappa\) of the matrix \(\mathbf{A}\).
x_hat <- solve(A, b, tol = 10 ^ -100)
matrix(c(x, x_hat), ncol = 2)## [,1] [,2]
## [1,] 1 0.9999999970547685
## [2,] 2 2.0000002072554426
## [3,] 3 2.9999936456039413
## [4,] 4 4.0001144903312822
## [5,] 5 4.9986270919975846
## [6,] 6 6.0117291183057908
## [7,] 7 6.9255380661381221
## [8,] 8 8.3611226585083571
## [9,] 9 7.6371835237744330
## [10,] 10 14.0500643229149667
## [11,] 11 1.4570592216348093
## [12,] 12 29.8690428304739264
## [13,] 13 -13.5340119325833328
## [14,] 14 45.0273178676153734
## [15,] 15 -13.1988463508062992
## [16,] 16 35.4922664012786768
## [17,] 17 7.1055228812786311
## [18,] 18 21.4752947809665287
## [19,] 19 18.2458383386842904
## [20,] 20 20.0761428395727357References
Sauer, Timothy. 2011. Numerical Analysis. 2nd ed. USA: Addison-Wesley Publishing Company.