############################################################
# Illustrates use of some utility commands available
# on einstein (although 'paste' is a generic Unix command)
# which are useful for manipulating columns of numbers.
#
# (1) dvmesh: Generates uniform sequence of real numbers.
# Included here mostly as a mechanism to generate
# input for 'nf'. (Instructor-supplied C-program).
#
# (2) nf: Generalization of 'nth'. (See Course Notes for
# September 25th.) Filter which selects columns from
# standard input (assumed numeric), performs fairly
# general mathematical operations as needed, and outputs
# one or more columns of numbers on standard output.
# (Instructor supplied perl-script).
#
# (3) paste: Standard Unix facility for combining ('pasting')
# one or more file arguments.,
############################################################
############################################################
# Usage of 'dvmesh' is straightforward. The command
# generates a length 'n' sequence of real numbers, uniformly
# spaced, and ranging from 'xmin' to 'xmax'.
############################################################
einstein% dvmesh
usage: dvmesh <xmin> <xmax> <n > 0>
einstein% dvmesh 0.0 1.0 11
0.0000000000000000E+00
1.0000000000000001E-01
2.0000000000000001E-01
3.0000000000000004E-01
4.0000000000000002E-01
5.0000000000000000E-01
5.9999999999999998E-01
6.9999999999999996E-01
7.9999999999999993E-01
8.9999999999999991E-01
1.0000000000000000E+00
############################################################
# 'nf' accepts an arbitrary number of arguments, reads
# columns of numbers from standard input, then manipulates
# the input-columns and writes the results to standard
# output. Use the notation '_1', '_2' etc. to refer to
# the first, second etc. column. Usage is best demonstrated
# with some examples:
############################################################
einstein% nf
usage: nf <expr #> [<expr #> ...]
############################################################
# Compute x^2, x = 0.0, 0.1, ... 0.9, 1.0 and write
# (x, x^2) to standard output. Note use of single quotes
# around 2nd argument to 'nf' to inhibit shell-interpretation
# of multiplication operator '*'.
############################################################
einstein% dvmesh 0.0 1.0 11 | nf _1 '_1 * _1'
0.0000000000000000E+00 0
1.0000000000000001E-01 0.01
2.0000000000000001E-01 0.04
3.0000000000000004E-01 0.09
4.0000000000000002E-01 0.16
5.0000000000000000E-01 0.25
5.9999999999999998E-01 0.36
6.9999999999999996E-01 0.49
7.9999999999999993E-01 0.64
8.9999999999999991E-01 0.81
1.0000000000000000E+00 1
############################################################
# Repeat the calculation and redirect to a file 'squares'.
# Compute the cubes of the same x-values and redirect
# (x,x^3) to 'cubes'.
############################################################
einstein% dvmesh 0.0 1.0 11 | nf _1 '_1 * _1' > squares
einstein% cat squares
0.0000000000000000E+00 0
1.0000000000000001E-01 0.01
2.0000000000000001E-01 0.04
3.0000000000000004E-01 0.09
4.0000000000000002E-01 0.16
5.0000000000000000E-01 0.25
5.9999999999999998E-01 0.36
6.9999999999999996E-01 0.49
7.9999999999999993E-01 0.64
8.9999999999999991E-01 0.81
1.0000000000000000E+00 1
einstein% dvmesh 0.0 1.0 11 | nf _1 'pow(_1,3)' > cubes
einstein% cat cubes
0.0000000000000000E+00 0
1.0000000000000001E-01 0.001
2.0000000000000001E-01 0.008
3.0000000000000004E-01 0.027
4.0000000000000002E-01 0.064
5.0000000000000000E-01 0.125
5.9999999999999998E-01 0.216
6.9999999999999996E-01 0.343
7.9999999999999993E-01 0.512
8.9999999999999991E-01 0.729
1.0000000000000000E+00 1
############################################################
# 'nf' understands
#
# (A) The usual binary arithmetic operations: *, /, +, -,
# (B) Integer power function (uses repeated multiplies)
# ipow(ix,iy) = ix^iy
# (C) Real power function (uses logs and exponentiation)
# pow(x,y) = x^y (x must be postive-definite)
# (D) min() and max() of an arbitrary number of arguments
# (E) The usual suite of mathematical functions: sin, cos,
# tan, sinh, cosh, tanh, exp, log, abs, sqrt (inverse
# trig and hyperbolic function are currently *not*
# implemented.
############################################################
einstein% dvmesh 0.0 4.0 11 | nf _1 'sin(_1)' 'cos(_1)' \
? 'ipow(sin(_1),2) + ipow(cos(_1),2)'
0.0000000000000000E+00 0 1 1
4.0000000000000002E-01 0.389418342308651 0.921060994002885 1
8.0000000000000004E-01 0.717356090899523 0.696706709347165 1
1.2000000000000002E+00 0.932039085967226 0.362357754476673 1
1.6000000000000001E+00 0.999573603041505 -0.0291995223012889 1
2.0000000000000000E+00 0.909297426825682 -0.416146836547142 1
2.3999999999999999E+00 0.675463180551151 -0.737393715541246 1
2.7999999999999998E+00 0.334988150155905 -0.942222340668658 1
3.1999999999999997E+00 -0.0583741434275798 -0.998294775794753 1
3.5999999999999996E+00 -0.442520443294852 -0.896758416334147 1
4.0000000000000000E+00 -0.756802495307928 -0.653643620863612 1
############################################################
# 'paste': Combines files 'horizontally' in a straightforward
# fashion. Most useful for use with two or more files each
# of which contain one or more columns with, but
# which all contain the same number of lines (length of
# columns). Note that paste's output is to standard out.
############################################################
einstein% paste squares cubes
0.0000000000000000E+00 0 0.0000000000000000E+00 0
1.0000000000000001E-01 0.01 1.0000000000000001E-01 0.001
2.0000000000000001E-01 0.04 2.0000000000000001E-01 0.008
3.0000000000000004E-01 0.09 3.0000000000000004E-01 0.027
4.0000000000000002E-01 0.16 4.0000000000000002E-01 0.064
5.0000000000000000E-01 0.25 5.0000000000000000E-01 0.125
5.9999999999999998E-01 0.36 5.9999999999999998E-01 0.216
6.9999999999999996E-01 0.49 6.9999999999999996E-01 0.343
7.9999999999999993E-01 0.64 7.9999999999999993E-01 0.512
8.9999999999999991E-01 0.81 8.9999999999999991E-01 0.729
1.0000000000000000E+00 1 1.0000000000000000E+00 1
############################################################
# The above is probably not quite what we wanted. Use
# 'nf' (or 'nth') to get rid of third column. Note that
# 'nth' refers to columnm 1, 2 etc simply as '1', '2'.
############################################################
einstein% paste squares cubes | nf _1 _2 _4
0.0000000000000000E+00 0 0
1.0000000000000001E-01 0.01 0.001
2.0000000000000001E-01 0.04 0.008
3.0000000000000004E-01 0.09 0.027
4.0000000000000002E-01 0.16 0.064
5.0000000000000000E-01 0.25 0.125
5.9999999999999998E-01 0.36 0.216
6.9999999999999996E-01 0.49 0.343
7.9999999999999993E-01 0.64 0.512
8.9999999999999991E-01 0.81 0.729
1.0000000000000000E+00 1 1
einstein% paste squares cubes | nth 1 2 4
0.0000000000000000E+00 0 0
1.0000000000000001E-01 0.01 0.001
2.0000000000000001E-01 0.04 0.008
3.0000000000000004E-01 0.09 0.027
4.0000000000000002E-01 0.16 0.064
5.0000000000000000E-01 0.25 0.125
5.9999999999999998E-01 0.36 0.216
6.9999999999999996E-01 0.49 0.343
7.9999999999999993E-01 0.64 0.512
8.9999999999999991E-01 0.81 0.729
1.0000000000000000E+00 1 1