16.8. Math Commands

"Doing the numbers"

factor

Decompose an integer into prime factors.

bash$ factor 2741727417: 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 10000PRIME=0E_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 # 0else return $E_NOTPRIME # nullfi}echofor 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.echoexit

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.gzechoecho "Given the principal, interest rate, and term of a mortgage," echo "calculate the monthly payment." bottom=1.0echoecho -n "Enter principal (no commas) " read principalecho -n "Enter interest rate (percent) " # If 12%, enter "12", not ".12".read interest_recho -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 stvfixed 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=85PN=`basename "$0"` # Program nameVER=`echo '$Revision: 1.2 $' | cut -d' ' -f2` # ==> VER=1.2Usage () { echo "$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) decimalotherwise (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 decimaldec=`echo "ibase=$ibase; $number" | bc` # ==> 'bc' is calculator utility.case "$dec" in [0-9]*); # number ok *)continue; # error: ignoreesac# 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. shiftdoneif [ $# -gt 0 ]then PrintBases "$@" else# Read from stdin. while read line doPrintBases $line donefiexit

An alternate method of invoking bcinvolves using a heredocument embedded within a command substitutionblock. This is especially appropriate when a scriptneeds to pass a list of options and commands tobc.

variable=`bc << LIMIT_STRINGoptionsstatementsoperationsLIMIT_STRING`...or...variable=$(bc << LIMIT_STRINGoptionsstatementsoperationsLIMIT_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 << EOF18.33 * 19.78EOF`echo $var1 # 362.56# $( ... ) notation also works.v1=23.53v2=17.881v3=83.501v4=171.63var2=$(bc << EOFscale = 4a = ( $v1 + $v2 )b = ( $v3 * $v4 )a * b + 15.35EOF)echo $var2 # 593487.8452var3=$(bc -l << EOFscale = 9s ( 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 << EOFscale = 9sqrt ( $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" # 8.184039344exit 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 << EOFscale = 0sqrt ( $1 * $1 + $2 * $2 )EOF)# Setting "scale" to zero rounds down result to integer value,#+ a necessary compromise in this script.# It decreases the accuracy of this simulation.}# ==========================================================# main() {# "Main" code block, mimicking a C-language main() function.# Initialize variables.shots=0splashes=0thuds=0Pi=0error=0while [ "$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" echodoneechoecho "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 crypticoperators. 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_NOARGSfi # Exercise: add argument validity checking.hexcvt (){if [ -z "$1" ]then echo 0 return # "Return" 0 if no arg passed to function.fiecho ""$1" "$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 fordc is a painful path to understanding itsintricacies. There seems to be a small, select group ofdc wizards who delight in showing offtheir mastery of this powerful, but arcane utility.

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

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 numberMIN=2 # Will not work for number smaller than this.E_NOARGS=85E_TOOSMALL=86if [ -z $1 ]then echo "Usage: $0 number" exit $E_NOARGSfiif [ "$1" -lt "$MIN" ]then echo "Number to factor must be $MIN or greater." exit $E_TOOSMALLfi # 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_BADARGSfiAWKSCRIPT=' { printf( "%3.7f", 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?