Linux, as with almost all UNIX systems, contains a vast number of little utilities tucked away in such places as /usr/bin and /usr/local/bin. One of these is the GNU utility bc.

bc is an arbitrary precision calculator language. It can perform arithmetic (both integer and real) to arbitrary precision, and it supports simple programming. It is started by the command:

bc -l files

The optional -l flag loads a mathematics library, and files (also optional) is a list of files containing bc commands. There are some other flags, but they do not greatly change the functionality. The mathematics library makes the following functions available to bc:

Let's look at a few examples of bc in action, assuming it has been started with the -l flag:

2^400
2582249878086908589655919172003011874329705792829\
2235128306593565406476220168411946296453532801378\
31435903171972747493376
scale=50
pi=4*a(1)
e(pi*sqrt(163))
262537412640768743.999999999999250072597198185688\
78393709875517366778
scale=100
l(2)
.693147180559945309417232121458176568075500134360\
2552541206800094933936219696947156058633269964186\
875

The value scale is one of bc's internal variables: it gives the number of figures to the right of the decimal point. Other versions of bc do not allow arbitrary values for scale. We could easily use 1000 instead of 10 in the following example, if we wanted more decimal places.

scale=10
4*a(1)
3.1415926532

bc provides most of the standard arithmetic operations:

scale=0
920^17%2773
948
.^157%2773
920

The period (.) is shorthand for the last result. The percentage sign % is the remainder function; it produces the standard integer remainder if scale is set to zero. When bc is invoked with the -l flag, the value of scale is set to 20.

Statements in bc are computed as quickly as possible. Thus, when using bc interactively, as shown above, statements are evaluated as soon as they are typed. A program in bc is simply a list of statements to be evaluated. The programming language provides loops, branches and recursion, and its syntax is similar to that of C. A simple example (from the man page) is the factorial function:

define f(x) {
if (x <= 1) return (1);
return (x*f(x-1));
}

It is convenient to place such definitions in a file (called, say things.b), and read them into bc with the command:

bc -l things.b

f(150)

5713383956445854590478932865261054003189553578601\
1264182548375833179829124845398393126574488675311\
1453771078787468542041626662501986845044663559491\
9592206657494259209573577892932535729044496247240\
5416790722118445437122269675520000000000000000000\
000000000000000000

define b1(n,k) {
if (k==0 || k==n) return (1);
return (b1(n-1,k)+b1(n-1,k-1));
}

b1(20,10)
184756

define b2(n,k) {
auto temp
temp=1;
if (k==0) return (1);
for(i=1; i<=k; i++) temp=temp*(n+1-i)/i;
return (temp);
}

h[1]=1
h[2]=1
for (i=3;i<=100;i++)
h[i]=h[i-h[i-1]]+h[i-h[i-2]]
h[10]
6
h[50]
25

for (i=1; i<=100; i++) {
print h[i],"       ";
if (i%10==0) print "\n;"
}
1    1    2    3    3    4    5    5    6    6
6    8    8    8    10   9    10   11   11   12
12   12   12   16   14   14   16   16   16   16
20   17   17   20   21   19   20   22   21   22
23   23   24   24   24   24   24   32   24   25
30   28   26   30   30   28   32   30   32   32
32   32   40   33   31   38   35   33   39   40
37   38   40   39   40   39   42   40   41   43
44   43   43   46   44   45   47   47   46   48
48   48   48   48   48   64   41   52   54   56

scale=0
define sf(n) {
auto sum,s;
sum=1;
s=sqrt(n);
for (i=2;i<=s;i++)
  if (n%i==0) sum=sum+i+n/i;
if (s*s==n) sum=sum-s;
return (sum);
}
define amicable(m) {
for (j=1;j<=m;j++)
  if (sf(sf(j))==j && sf(j)!=j && j<sf(j)) print
        j,"        ",sf(j),"\n";
print "Done.\n";
}

Second, Simpson's rule for numerical integration:

define simpson(a,b,n) {
auto h,sum_even,sum_odd;
h=(b-a)/(2*n);
sum_even=0;
sum_odd=0;
for (i=1;i<=n;i++) sum_odd=sum_odd+f(a+(2*i-1)*h);
for(i=1;i<n;i++) sum_even=sum_even+f(a+2*i*h);
return ((f(a)+f(b)+4*sum_odd+2*sum_even)*h/3);
}

Defining a function f(x) by, say:

define f(x) {
return (e(-(x^2)));
}

simpson(0,1,10)

In my opinion, bc is a real find: it is small, efficient, self-contained and an extremely useful utility. It is not to be considered a replacement for a good fast programming language such as C, C++ or FORTRAN. But as a means for quickly prototyping numerical algorithms before coding them in a high-level language, it is excellent.

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