Math Commands

Advanced Bash-Scripting Guide:
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# 16.8. Math Commands

"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

      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 (heiner.steven@odn.de)
      # 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 <charpov@cs.unh.edu>
      #  (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?

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