15.8. Math Commands

"Doing the numbers"

factor

Decompose an integer into prime factors.

bash$ factor 27417 27417: 3 13 19 37

Example 15-46. 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 comes to the rescue.

Not just a versatile, arbitrary precision calculation utility, bc offers many of the facilities of a programming language.

bc 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 15-47. 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 author of the ABS Guide). # http://www.ibiblio.org/pub/Linux/apps/financial/mcalc-1.6.tar.gz [15k] 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 15-48. 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 15-49. 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 15-50. 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 reason for this 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. # (The previous explanation was a simplification.) # # 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 a few compromises. # This tends to lower 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 to approximate PI. 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 * $2 ) EOF ) # Setting "scale" to zero rounds down result to integer value, #+ a necessary compromise in this script. # This decreases the accuracy of the simulation. } # ========================================================== # main() { # "Main" code block, mimmicking a C-language main() function. # Initialize variables. shots=0 splashes=0 thuds=0 Pi=0 error=0 while [ "$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, #+ probably due to round-off error and imperfect randomness of $RANDOM. # But still usually within plus-or-minus 5% . . . #+ a pretty good rough approximation. error=$(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 # 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.

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 15-51. 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 15-52. 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 15-53. 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?