16.8. Math Commands (Revision #3)

"Doing the numbers"

factor

Decompose an integer into prime factors.

bash$ factor 27417
27417: 3 13 19 37
      

Example 16-46. Generating prime numbers

#!/bin/bash
# primes2.sh

#  Generating prime numbers the quick-and-easy way,
#+ without resorting to fancy algorithms.

CEILING=10000   # 1 to 10000
PRIME=0
E_NOTPRIME=

is_prime ()
{
  local factors
  factors=( $(factor $1) )  # Load output of `factor` into array.

if [ -z "${factors[2]}" ]
#  Third element of "factors" array:
#+ ${factors[2]} is 2nd factor of argument.
#  If it is blank, then there is no 2nd factor,
#+ and the argument is therefore prime.
then
  return $PRIME             # 0
else
  return $E_NOTPRIME        # null
fi
}

echo
for n in $(seq $CEILING)
do
  if is_prime $n
  then
    printf %5d $n
  fi   #    ^  Five positions per number suffices.
done   #       For a higher $CEILING, adjust upward, as necessary.

echo

exit

bc

Bash can't handle floating point calculations, and it lacks operators for certain important mathematical functions. Fortunately, bc gallops to the rescue.

Not just a versatile, arbitrary precision calculation utility, bc offers many of the facilities of a programming language. It has a syntax vaguely resembling C.

Since it is a fairly well-behaved UNIX utility, and may therefore be used in a pipe, bc comes in handy in scripts.

Here is a simple template for using bc to calculate a script variable. This uses command substitution.

                    variable=$(echo "OPTIONS; OPERATIONS" | bc)

Example 16-47. Monthly Payment on a Mortgage

#!/bin/bash
# monthlypmt.sh: Calculates monthly payment on a mortgage.

#  This is a modification of code in the
#+ "mcalc" (mortgage calculator) package,
#+ by Jeff Schmidt
#+ and
#+ Mendel Cooper (yours truly, the ABS Guide author).
#  http://www.ibiblio.org/pub/Linux/apps/financial/mcalc-1.6.tar.gz">http://www.ibiblio.org/pub/Linux/apps/financial/mcalc-1.6.tar.gz

echo
echo "Given the principal, interest rate, and term of a mortgage,"
echo "calculate the monthly payment."

bottom=1.0

echo
echo -n "Enter principal (no commas) "
read principal
echo -n "Enter interest rate (percent) "  # If 12%, enter "12", not ".12".
read interest_r
echo -n "Enter term (months) "
read term

 interest_r=$(echo "scale=9; $interest_r/100.0" | bc) # Convert to decimal.
                 #           ^^^^^^^^^^^^^^^^^  Divide by 100.
                 # "scale" determines how many decimal places.

 interest_rate=$(echo "scale=9; $interest_r/12 + 1.0" | bc)

 top=$(echo "scale=9; $principal*$interest_rate^$term" | bc)
          #           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
          #           Standard formula for figuring interest.

 echo; echo "Please be patient. This may take a while."

 let "months = $term - 1"
# ====================================================================
 for ((x=$months; x > 0; x--))
 do
   bot=$(echo "scale=9; $interest_rate^$x" | bc)
   bottom=$(echo "scale=9; $bottom+$bot" | bc)
#  bottom = $(($bottom + $bot"))
 done
# ====================================================================

# --------------------------------------------------------------------
#  Rick Boivie pointed out a more efficient implementation
#+ of the above loop, which decreases computation time by 2/3.

# for ((x=1; x <= $months; x++))
# do
#   bottom=$(echo "scale=9; $bottom * $interest_rate + 1" | bc)
# done

#  And then he came up with an even more efficient alternative,
#+ one that cuts down the run time by about 95%!

# bottom=`{
#     echo "scale=9; bottom=$bottom; interest_rate=$interest_rate"
#     for ((x=1; x <= $months; x++))
#     do
#          echo 'bottom = bottom * interest_rate + 1'
#     done
#     echo 'bottom'
#     } | bc`       # Embeds a 'for loop' within command substitution.
# --------------------------------------------------------------------------
#  On the other hand, Frank Wang suggests:
#  bottom=$(echo "scale=9; ($interest_rate^$term-1)/($interest_rate-1)" | bc)

#  Because . . .
#  The algorithm behind the loop
#+ is actually a sum of geometric proportion series.
#  The sum formula is e0(1-q^n)/(1-q),
#+ where e0 is the first element and q=e(n+1)/e(n)
#+ and n is the number of elements.
# --------------------------------------------------------------------------

 # let "payment = $top/$bottom"
 payment=$(echo "scale=2; $top/$bottom" | bc)
 # Use two decimal places for dollars and cents.
 
 echo
 echo "monthly payment = \$$payment"  # Echo a dollar sign in front of amount.
 echo

 exit 0

 # Exercises:
 #   1) Filter input to permit commas in principal amount.
 #   2) Filter input to permit interest to be entered as percent or decimal.
 #   3) If you are really ambitious,
 #+     expand this script to print complete amortization tables.

Example 16-48. Base Conversion

#!/bin/bash
###########################################################################
# Shellscript: base.sh - print number to different bases (Bourne Shell)
# Author     : Heiner Steven ([email protected])
# Date       : 07-03-95
# Category   : Desktop
# $Id: base.sh,v 1.2 2000/02/06 19:55:35 heiner Exp $
# ==> Above line is RCS ID info.
###########################################################################
# Description
#
# Changes
# 21-03-95 stv fixed error occuring with 0xb as input (0.2)
###########################################################################

# ==> Used in ABS Guide with the script author's permission.
# ==> Comments added by ABS Guide author.

NOARGS=85
PN=`basename "$0"`          # Program name
VER=`echo '$Revision: 1.2 $' | cut -d' ' -f2`  # ==> VER=1.2

Usage () {
    echo "$PN - print number to different bases, $VER (stv '95)
usage: $PN [number ...]

If no number is given, the numbers are read from standard input.
A number may be
    binary (base 2)  starting with 0b (i.e. 0b1100)
    octal (base 8)  starting with 0  (i.e. 014)
    hexadecimal (base 16) starting with 0x (i.e. 0xc)
    decimal   otherwise (i.e. 12)" >&2
    exit $NOARGS
}   # ==> Prints usage message.

Msg () {
    for i   # ==> in [list] missing. Why?
    do echo "$PN: $i" >&2
    done
}

Fatal () { Msg "$@"; exit 66; }

PrintBases () {
    # Determine base of the number
    for i      # ==> in [list] missing...
    do         # ==> so operates on command-line arg(s).
 case "$i" in
     0b*)  ibase=2;; # binary
     0x*|[a-f]*|[A-F]*) ibase=16;; # hexadecimal
     0*)   ibase=8;; # octal
     [1-9]*)  ibase=10;; # decimal
     *)
  Msg "illegal number $i - ignored"
  continue;;
 esac

 # Remove prefix, convert hex digits to uppercase (bc needs this).
 number=`echo "$i" | sed -e 's:^0[bBxX]::' | tr '[a-f]' '[A-F]'`
 # ==> Uses ":" as sed separator, rather than "/".

 # Convert number to decimal
 dec=`echo "ibase=$ibase; $number" | bc`  # ==> 'bc' is calculator utility.
 case "$dec" in
     [0-9]*) ;;    # number ok
     *)  continue;;   # error: ignore
 esac

 # Print all conversions in one line.
 # ==> 'here document' feeds command list to 'bc'.
 echo `bc <<!
     obase=16; "hex="; $dec
     obase=10; "dec="; $dec
     obase=8;  "oct="; $dec
     obase=2;  "bin="; $dec
!
    ` | sed -e 's: : :g'

    done
}

while [ $# -gt 0 ]
# ==>  Is a "while loop" really necessary here,
# ==>+ since all the cases either break out of the loop
# ==>+ or terminate the script.
# ==> (Above comment by Paulo Marcel Coelho Aragao.)
do
    case "$1" in
 --)     shift; break;;
 -h)     Usage;;                 # ==> Help message.
 -*)     Usage;;
         *)     break;;                 # First number
    esac   # ==> Error checking for illegal input might be appropriate.
    shift
done

if [ $# -gt 0 ]
then
    PrintBases "$@"
else     # Read from stdin.
    while read line
    do
 PrintBases $line
    done
fi

exit

An alternate method of invoking bc involves using a here document embedded within a command substitution block. This is especially appropriate when a script needs to pass a list of options and commands to bc.

variable=`bc << LIMIT_STRING
options
statements
operations
LIMIT_STRING
`

...or...

variable=$(bc << LIMIT_STRING
options
statements
operations
LIMIT_STRING
)

Example 16-49. Invoking bc using a here document

#!/bin/bash
# Invoking 'bc' using command substitution
# in combination with a 'here document'.

var1=`bc << EOF
18.33 * 19.78
EOF
`
echo $var1       # 362.56

#  $( ... ) notation also works.
v1=23.53
v2=17.881
v3=83.501
v4=171.63

var2=$(bc << EOF
scale = 4
a = ( $v1 + $v2 )
b = ( $v3 * $v4 )
a * b + 15.35
EOF
)
echo $var2       # 593487.8452

var3=$(bc -l << EOF
scale = 9
s ( 1.7 )
EOF
)
# Returns the sine of 1.7 radians.
# The "-l" option calls the 'bc' math library.
echo $var3       # .991664810

# Now, try it in a function...
hypotenuse ()    # Calculate hypotenuse of a right triangle.
{                # c = sqrt( a^2 + b^2 )
hyp=$(bc -l << EOF
scale = 9
sqrt ( $1 * $1 + $2 * $2 )
EOF
)
# Can't directly return floating point values from a Bash function.
# But, can echo-and-capture:
echo "$hyp"
}

hyp=$(hypotenuse 3.68 7.31)
echo "hypotenuse = $hyp"    # 8.184039344

exit 0

Example 16-50. Calculating PI

#!/bin/bash
# cannon.sh: Approximating PI by firing cannonballs.

# Author: Mendel Cooper
# License: Public Domain
# Version 2.2, reldate 13oct08.

# This is a very simple instance of a "Monte Carlo" simulation:
#+ a mathematical model of a real-life event,
#+ using pseudorandom numbers to emulate random chance.

#  Consider a perfectly square plot of land, 10000 units on a side.
#  This land has a perfectly circular lake in its center,
#+ with a diameter of 10000 units.
#  The plot is actually mostly water, except for land in the four corners.
#  (Think of it as a square with an inscribed circle.)
#
#  We will fire iron cannonballs from an old-style cannon
#+ at the square.
#  All the shots impact somewhere on the square,
#+ either in the lake or on the dry corners.
#  Since the lake takes up most of the area,
#+ most of the shots will SPLASH! into the water.
#  Just a few shots will THUD! into solid ground
#+ in the four corners of the square.
#
#  If we take enough random, unaimed shots at the square,
#+ Then the ratio of SPLASHES to total shots will approximate
#+ the value of PI/4.
#
#  The simplified explanation is that the cannon is actually
#+ shooting only at the upper right-hand quadrant of the square,
#+ i.e., Quadrant I of the Cartesian coordinate plane.
#
#
#  Theoretically, the more shots taken, the better the fit.
#  However, a shell script, as opposed to a compiled language
#+ with floating-point math built in, requires some compromises.
#  This decreases the accuracy of the simulation.

DIMENSION=10000  # Length of each side of the plot.
                 # Also sets ceiling for random integers generated.

MAXSHOTS=1000    # Fire this many shots.
                 # 10000 or more would be better, but would take too long.
PMULTIPLIER=4.0  # Scaling factor.

declare -r M_PI=3.141592654
                 # Actual 9-place value of PI, for comparison purposes.

get_random ()
{
SEED=$(head -n 1 /dev/urandom | od -N 1 | awk '{ print $2 }')
RANDOM=$SEED                                  #  From "seeding-random.sh"
                                              #+ example script.
let "rnum = $RANDOM % $DIMENSION"             #  Range less than 10000.
echo $rnum
}

distance=        # Declare global variable.
hypotenuse ()    # Calculate hypotenuse of a right triangle.
{                # From "alt-bc.sh" example.
distance=$(bc -l << EOF
scale = 0
sqrt ( $1 * $1 + $2 * $2 )
EOF
)
#  Setting "scale" to zero rounds down result to integer value,
#+ a necessary compromise in this script.
#  It decreases the accuracy of this simulation.
}

# ==========================================================
# main() {
# "Main" code block, mimicking a C-language main() function.

# Initialize variables.
shots=0
splashes=0
thuds=0
Pi=0
error=0

while [ "$shots" -lt  "$MAXSHOTS" ]           # Main loop.
do

  xCoord=$(get_random)                        # Get random X and Y coords.
  yCoord=$(get_random)
  hypotenuse $xCoord $yCoord                  #  Hypotenuse of
                                              #+ right-triangle = distance.
  ((shots++))

  printf "#%4d   " $shots
  printf "Xc = %4d  " $xCoord
  printf "Yc = %4d  " $yCoord
  printf "Distance = %5d  " $distance         #   Distance from
                                              #+  center of lake
                                              #+  -- the "origin" --
                                              #+  coordinate (0,0).

  if [ "$distance" -le "$DIMENSION" ]
  then
    echo -n "SPLASH!  "
    ((splashes++))
  else
    echo -n "THUD!    "
    ((thuds++))
  fi

  Pi=$(echo "scale=9; $PMULTIPLIER*$splashes/$shots" | bc)
  # Multiply ratio by 4.0.
  echo -n "PI ~ $Pi"
  echo

done

echo
echo "After $shots shots, PI looks like approximately   $Pi"
#  Tends to run a bit high,
#+ possibly due to round-off error and imperfect randomness of $RANDOM.
#  But still usually within plus-or-minus 5% . . .
#+ a pretty fair rough approximation.
error=$(echo "scale=9; $Pi - $M_PI" | bc)
pct_error=$(echo "scale=2; 100.0 * $error / $M_PI" | bc)
echo -n "Deviation from mathematical value of PI =        $error"
echo " ($pct_error% error)"
echo

# End of "main" code block.
# }
# ==========================================================

exit 0

#  One might well wonder whether a shell script is appropriate for
#+ an application as complex and computation-intensive as a simulation.
#
#  There are at least two justifications.
#  1) As a proof of concept: to show it can be done.
#  2) To prototype and test the algorithms before rewriting
#+    it in a compiled high-level language.

See also Example A-37.

dc

The dc (desk calculator) utility is stack-oriented and uses RPN (Reverse Polish Notation). Like bc, it has much of the power of a programming language.

Similar to the procedure with bc, echo a command-string to dc.

echo "[Printing a string ... ]P" | dc
# The P command prints the string between the preceding brackets.

# And now for some simple arithmetic.
echo "7 8 * p" | dc     # 56
#  Pushes 7, then 8 onto the stack,
#+ multiplies ("*" operator), then prints the result ("p" operator).

Most persons avoid dc, because of its non-intuitive input and rather cryptic operators. Yet, it has its uses.

Example 16-51. Converting a decimal number to hexadecimal

#!/bin/bash
# hexconvert.sh: Convert a decimal number to hexadecimal.

E_NOARGS=85 # Command-line arg missing.
BASE=16     # Hexadecimal.

if [ -z "$1" ]
then        # Need a command-line argument.
  echo "Usage: $0 number"
  exit $E_NOARGS
fi          # Exercise: add argument validity checking.

hexcvt ()
{
if [ -z "$1" ]
then
  echo 0
  return    # "Return" 0 if no arg passed to function.
fi

echo ""$1" "$BASE" o p" | dc
#                  o    sets radix (numerical base) of output.
#                    p  prints the top of stack.
# For other options: 'man dc' ...
return
}

hexcvt "$1"

exit

Studying the info page for dc is a painful path to understanding its intricacies. There seems to be a small, select group of dc wizards who delight in showing off their mastery of this powerful, but arcane utility.

bash$ echo "16i[q]sa[ln0=aln100%Pln100/snlbx]sbA0D68736142snlbxq" | dc
Bash

      

dc <<< 10k5v1+2/p # 1.6180339887
#  ^^^            Feed operations to dc using a Here String.
#      ^^^        Pushes 10 and sets that as the precision (10k).
#         ^^      Pushes 5 and takes its square root
#                 (5v, v = square root).
#           ^^    Pushes 1 and adds it to the running total (1+).
#             ^^  Pushes 2 and divides the running total by that (2/).
#               ^ Pops and prints the result (p)
#  The result is  1.6180339887 ...
#  ... which happens to be the Pythagorean Golden Ratio, to 10 places.

Example 16-52. Factoring

#!/bin/bash
# factr.sh: Factor a number

MIN=2       # Will not work for number smaller than this.
E_NOARGS=85
E_TOOSMALL=86

if [ -z $1 ]
then
  echo "Usage: $0 number"
  exit $E_NOARGS
fi

if [ "$1" -lt "$MIN" ]
then
  echo "Number to factor must be $MIN or greater."
  exit $E_TOOSMALL
fi 

# Exercise: Add type checking (to reject non-integer arg).

echo "Factors of $1:"
# -------------------------------------------------------
echo  "$1[p]s2[lip/dli%0=1dvsr]s12sid2%0=13sidvsr[dli%0=\
1lrli2+dsi!>.]ds.xd1<2" | dc
# -------------------------------------------------------
#  Above code written by Michel Charpentier <[email protected]>
#  (as a one-liner, here broken into two lines for display purposes).
#  Used in ABS Guide with permission (thanks!).

 exit

 # $ sh factr.sh 270138
 # 2
 # 3
 # 11
 # 4093

awk

Yet another way of doing floating point math in a script is using awk's built-in math functions in a shell wrapper.

Example 16-53. Calculating the hypotenuse of a triangle

#!/bin/bash
# hypotenuse.sh: Returns the "hypotenuse" of a right triangle.
#                (square root of sum of squares of the "legs")

ARGS=2                # Script needs sides of triangle passed.
E_BADARGS=85          # Wrong number of arguments.

if [ $# -ne "$ARGS" ] # Test number of arguments to script.
then
  echo "Usage: `basename $0` side_1 side_2"
  exit $E_BADARGS
fi

AWKSCRIPT=' { printf( "%3.7f\n", sqrt($1*$1 + $2*$2) ) } '
#             command(s) / parameters passed to awk

# Now, pipe the parameters to awk.
    echo -n "Hypotenuse of $1 and $2 = "
    echo $1 $2 | awk "$AWKSCRIPT"
#   ^^^^^^^^^^^^
# An echo-and-pipe is an easy way of passing shell parameters to awk.

exit

# Exercise: Rewrite this script using 'bc' rather than awk.
#           Which method is more intuitive?