Note: If you wish to replicate the R code below, then you will need to copy and paste the following commands in R first (to make sure you have all the packages and their dependencies):

install.packages("install.load")
# install the install.load package

# you will need to have GNU Octave (check the RcppOctave CRAN page for more details) installed on your system, then you can use the code below:

install.load::install_package("ramify", "ggplot2", "ggpmisc", "RcppOctave")
# install and/or load the packages and their dependencies, including the extra system dependencies (this process may take a while depending on the number of dependencies)


install.packages("import")
# install the import package

import::from(pracma, golden_ratio, fminbnd)
# import golden_ratio, fminbnd from the pracma package


Example 1 (modified from Source 1 and Wickham)



# Source 1 begins

# option 1
integrate_vector <- Vectorize(integrate, vectorize.args = c("f", "lower", "upper"))

funcs <- list(function(x) x^5 - 7 * x^4 + 9 * x^2 - 3 * x + 1, function(x) 8 * 
    sin(x) + 3 * cos(x), function(x) 8 * exp(x)/x^4)

integrate_vector(funcs, lower = c(0, -4 * pi, 5), upper = c(100, 4 * pi, 30))
##              [,1]         [,2]         [,3]      
## value        152669651767 -5.58059e-15 122584887 
## abs.error    0.001694974  3.323246e-08 128.3941  
## subdivisions 1            2            1         
## message      "OK"         "OK"         "OK"      
## call         Expression   Expression   Expression
# option 2
f1 <- c(f = function(x) x^5 - 7 * x^4 + 9 * x^2 - 3 * x + 1, l = 0, u = 100)

f2 <- c(f = function(x) 8 * sin(x) + 3 * cos(x), l = -4 * pi, u = 4 * pi)

f3 <- c(f = function(x) 8 * exp(x)/x^4, l = 5, u = 30)

funs <- list(f1, f2, f3)

lapply(funs, function(x) integrate(x[["f"]], x[["l"]], x[["u"]]))
## [[1]]
## 152669651767 with absolute error < 0.0017
## 
## [[2]]
## -5.58059e-15 with absolute error < 3.3e-08
## 
## [[3]]
## 122584887 with absolute error < 128
# option 3
functions <- list(function(x) x^5 - 7 * x^4 + 9 * x^2 - 3 * x + 1, function(x) 8 * 
    sin(x) + 3 * cos(x), function(x) 8 * exp(x)/x^4)

lower <- c(0, -4 * pi, 5)

upper <- c(100, 4 * pi, 30)

mapply(integrate, functions, lower, upper)
##              [,1]         [,2]         [,3]      
## value        152669651767 -5.58059e-15 122584887 
## abs.error    0.001694974  3.323246e-08 128.3941  
## subdivisions 1            2            1         
## message      "OK"         "OK"         "OK"      
## call         Expression   Expression   Expression
# Source 1 ends



Example 2

“Suppose you have measured temperatures at a number of coordinates on the surface of a rectangular heated plate”:

“T(2, 1) = 60
T(9, 1) = 57.5
T(2, 6) = 55
T(9, 6) = 70”

“Use bilinear interpolation to estimate the temperature at xi = 5.25 and yi = 4.8.” (Chapra 381)



library(ramify)

import::from(pracma, interp2)
# import interp2 from the pracma package


x <- mat("2, 9")
x
##      [,1] [,2]
## [1,]    2    9
y <- mat("1, 6")
y
##      [,1] [,2]
## [1,]    1    6
z <- mat("60, 57.5; 55, 70")
z
##      [,1] [,2]
## [1,]   60 57.5
## [2,]   55 70.0
T <- interp2(x, y, z, 5.25, 4.8)

T
## [1] 61.21429

The “temperature at xi = 5.25 and yi = 4.8” is 61."



Using GNU Octave through RcppOctave to obtain the answer for Chapra 381:



library(RcppOctave)

o_source(text = "
x = [2 9]

y = [1 6]

z = [60 57.5; 55 70]

interp2(x, y, z, 5.25, 4.8)
")
## x =
## 
##    2   9
## 
## y =
## 
##    1   6
## 
## z =
## 
##    60.000   57.500
##    55.000   70.000
## 
## ans =  61.214



Example 3

“Determine the point of maximum deflection of a beam using the golden-section search.” (Chapra 184) / Problem 7.16



install.load::load_package("ggplot2", "ggpmisc")
# load needed packages using the load_package function from the install.load
# package (it is assumed that you have already installed these packages)

import::from(pracma, golden_ratio)
# import golden_ratio from the pracma package


source("https://raw.githubusercontent.com/ggrothendieck/gsubfn/master/R/list.R")


y <- function(x) {
    y1 <- (w0/(120 * E * I * L)) * (-x^5 + 2 * L^2 * x^3 - L^4 * x)
    return(y1)
}

L <- 600  # cm

E <- 50000  # kN/cm^2

I <- 30000  # cm^2

w0 <- 2.5  # kN/cm

epsilons <- 1/100  # %

xl <- 0

xu <- L

list[xmin, fmin, iter, ea] <- golden_ratio(y, xl, xu, tol = epsilons)

xmin
## [1] 268.3303
fmin
## [1] -0.5151901
iter
## [1] 20
ea
## [1] 0.009363442
# use ggplot2 to create the figure
ggplot(data.frame(x = xl:xu, y = y(xl:xu)), aes(x, y)) + geom_line() + labs(x = "Length (cm)", 
    y = "Deflection (cm)", title = "Beam Deflection") + stat_valleys(colour = "purple") + 
    stat_valleys(y.label.fmt = "%1.2f", geom = "text", colour = "red", vjust = 1.5, 
        aes(label = paste(..y.label.., "cm at", ..x.label.., "cm"))) + theme_bw()  # Source 2



Using GNU Octave through RcppOctave to obtain the answer for Chapra 184 / Problem 7.16:



library(RcppOctave)

o_source(text = "
L = 600 % cm

E = 50000 % kN/cm^2

I = 30000 % cm^2

w0 = 2.5 % kN/cm

epsilons = 1 / 100 % %

xl = 0

xu = L


z = @(x) (w0 / (120 * E * I * L)) * (-x ^ 5 + 2 * L ^ 2 * x ^ 3 - L ^ 4 * x)


function [x, k] = golden_section( f, a, b, tol, maxiter, verbose )
  %---------------------------------------------------------------------------
  % Jonathan R. Senning <jonathan.senning@gordon.edu>
  % Gordon College
  % Written May 3, 1999
  % Revised May 4, 2005
  % Revised December 18, 2008 - Works with both Matlab and Octave
  %
  % Implements the golden section search as described on pages 555-556 of
  % Numerical Mathematics and Computing, 4th Edition, by Cheney & Kincaid,
  % Brooks/Cole, 1999 except that y is calculated as an offset from b rather
  % than from a.
  %---------------------------------------------------------------------------

  if ( nargin <= 5 )
    verbose = 0;
  end

  % Set things up to begin the iteration
  r  = ( sqrt( 5 ) - 1 ) / 2;   % Golden ratio
  x = a + r * ( b - a );
  y = b - r * ( b - a );
  u = f( x );
  v = f( y );

  % Main loop
  k = 0;
  if ( verbose ~= 0 )
    fprintf( '%4d: Interval is [%f, %f]; y = %f, x = %f\n', k, a, b, y, x );
  end
  while ( abs( b - a ) >= 2 * tol && k < maxiter )
    k = k + 1;
    if ( u > v )        % minimum must be in [a, x]
      b = x;
      x = y;
      u = v;
      y = b - r * ( b - a );
      v = f( y );
    else                % minimum must be in [y, b]
      a = y;
      y = x;
      v = u;
      x = a + r * ( b - a );
      u = f( x );
    end
    if ( verbose ~= 0 )
      fprintf( '%4d: Interval is [%f, %f]; y = %f, x = %f\n', k, a, b, y, x );
    end
  end

  % All done; best estimate of solution is midpoint of final interval
  x = 0.5 * ( a + b );

endfunction % needed for GNU Octave only



[xmin, iter] = golden_section(z, 0, 8, tol = 1e-08, maxiter = 100)


xmin

iter
")



Example 4

“Determine the critical travel time and concentration” using the Streeter-Phelps model. (Chapra 185) / Problem 7.25



import::from(pracma, fminbnd)
# import fminbnd from the pracma package


source("https://raw.githubusercontent.com/ggrothendieck/gsubfn/master/R/list.R")

o <- function(t) {
    o1 <- os - ((kd * L0/(kd + ks - ka)) * (exp(-ka * t) - exp((-kd + ks) * 
        t))) - Sb/ka * (1 - exp(-ka * t))
    return(o1)
}

os <- 10  # mg/L

kd <- 0.1  # d^-1

ka <- 0.6  # d^-1

ks <- 0.05  # d^-1

L0 <- 50  # mg/L

Sb <- 1  # mg/L/d

list[oc, tc] <- fminbnd(o, 0, 5)

oc
## [1] 4.772125
tc
## [1] 0.3102083



Using GNU Octave through RcppOctave to obtain the answer for Chapra 184 / Problem 7.16:



library(RcppOctave)

o_source(text = "
function o1 = o(t)
global os kd L0 ks ka Sb
o1 = os - ((kd * L0 / (kd + ks - ka)) * (exp(-ka * t) - exp((-kd + ks) * t))) - Sb / ka * (1 - exp(-ka * t));
endfunction % only needed for GNU Octave


global os kd L0 ks ka Sb
os = 10 % mg/L

kd = 0.1 % d^-1

ka = 0.6 % d^-1

ks = 0.05 % d^-1

L0 = 50 % mg/L

Sb = 1 % mg/L/d

[oc, tc] = fminbnd(@o, 0, 5)

oc

tc
")
## os =  10
## kd =  0.10000
## ka =  0.60000
## ks =  0.050000
## L0 =  50
## Sb =  1
## oc =  4.7721
## tc =  0.31021
## oc =  4.7721
## tc =  0.31021



Sources used in the R code

Source 1
anonymous function exercise in Wickham’s Advanced R book (integration of 3 functions in list) - Stack Overflow answered by jlhoward on Dec 19 2014 & answered and edited by Khashaa on Dec 19 2014. See https://stackoverflow.com/questions/27572408/anonymous-function-exercise-in-wickhams-advanced-r-book.

Source 2
Using R for photobiology: R tips 3: ggpmisc adds new stats to ‘ggplot2’ by Pedro J. Aphalo, 2016-01-30. See http://www.r4photobiology.info/2016/01/ggpmisc-adds-new-stats-to-ggplot2/.



Works Cited

Hadley Wickham, Advanced R (Online), “Functional programming”, http://adv-r.had.co.nz/Functional-programming.html

Jonathan R. Senning, “golden_section GNU Octave/MATLAB program, Gordon College, Revised December 18, 2008, http://www.math-cs.gordon.edu/courses/mat342/octave/golden_section.m.

Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, 2nd Edition, Boston, Massachusetts: McGraw-Hill, 2008, pages 184-185, 381.



---
title: "Numerical Methods Examples with R"
author: "Irucka Embry, E.I.T."
date: "`r Sys.Date()`"
output:
html_document:
mathjax: default
---

<br />
<br />

Note: If you wish to replicate the R code below, then you will need to copy and paste the following commands in R first (to make sure you have all the packages and their dependencies):

```{r eval = FALSE}
install.packages("install.load")
# install the install.load package

# you will need to have GNU Octave (check the RcppOctave CRAN page for more details) installed on your system, then you can use the code below:

install.load::install_package("ramify", "ggplot2", "ggpmisc", "RcppOctave")
# install and/or load the packages and their dependencies, including the extra system dependencies (this process may take a while depending on the number of dependencies)


install.packages("import")
# install the import package

import::from(pracma, golden_ratio, fminbnd)
# import golden_ratio, fminbnd from the pracma package
```

<br />

# Example 1 (modified from Source 1 and Wickham)

<br />
<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
# Source 1 begins

# option 1
integrate_vector <- Vectorize(integrate, vectorize.args = c("f", "lower", "upper"))

funcs <- list(function (x) x ^ 5 - 7 * x ^ 4 + 9 * x ^ 2 - 3 * x + 1, function(x) 8 * sin(x) + 3 * cos(x), function(x) 8 * exp(x) / x ^ 4)

integrate_vector(funcs, lower = c(0, -4 * pi, 5), upper = c(100, 4 * pi, 30))


# option 2
f1 <- c(f = function(x) x ^ 5 - 7 * x ^ 4 + 9 * x ^ 2 - 3 * x + 1, l = 0, u = 100)

f2 <- c(f = function(x) 8 * sin(x) + 3 * cos(x), l = -4 * pi, u = 4 * pi)

f3 <- c(f = function(x) 8 * exp(x) / x ^ 4, l = 5, u = 30)

funs <- list(f1, f2, f3)

lapply(funs, function(x) integrate(x[["f"]], x[["l"]], x[["u"]]))


# option 3
functions <- list(function(x) x ^ 5 - 7 * x ^ 4 + 9 * x ^ 2 - 3 * x + 1, function(x) 8 * sin(x) + 3 * cos(x), function(x) 8 * exp(x) / x ^ 4)

lower <- c(0, -4 * pi, 5)

upper <- c(100, 4 * pi, 30)

mapply(integrate, functions, lower, upper)
# Source 1 ends
```

<br />
<br />

# Example 2

"Suppose you have measured temperatures at a number of coordinates on the surface of a rectangular heated plate":

"T(2, 1) = 60<br />
T(9, 1) = 57.5<br />
T(2, 6) = 55<br />
T(9, 6) = 70"<br />

"Use bilinear interpolation to estimate the temperature at x~*i*~ = 5.25 and y~*i*~ = 4.8." (Chapra 381)

<br />
<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
library(ramify)

import::from(pracma, interp2)
# import interp2 from the pracma package


x <- mat("2, 9"); x

y <- mat("1, 6"); y

z <- mat("60, 57.5; 55, 70"); z

T <- interp2(x, y, z, 5.25, 4.8)

T
```

The "temperature at x~*i*~ = 5.25 and y~*i*~ = 4.8" is `r round(T)`."

<br />
<br />

Using GNU Octave through RcppOctave to obtain the answer for Chapra 381:

<br />
<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
library(RcppOctave)

o_source(text = "
x = [2 9]

y = [1 6]

z = [60 57.5; 55 70]

interp2(x, y, z, 5.25, 4.8)
")
```

<br />
<br />

# Example 3

"Determine the point of maximum deflection of a beam using the golden-section search." (Chapra 184) / Problem 7.16

<br />
<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
install.load::load_package("ggplot2", "ggpmisc")
# load needed packages using the load_package function from the install.load package (it is assumed that you have already installed these packages)

import::from(pracma, golden_ratio)
# import golden_ratio from the pracma package


source("https://raw.githubusercontent.com/ggrothendieck/gsubfn/master/R/list.R")


y <- function(x) {
y1 <- (w0 / (120 * E * I * L)) * (-x ^ 5 + 2 * L ^ 2 * x ^ 3 - L ^ 4 * x)
return(y1)
}

L <- 600 # cm

E <- 50000 # kN/cm^2

I <- 30000 # cm^2

w0 <- 2.5 # kN/cm

epsilons <- 1 / 100 # %

xl <- 0

xu <- L

list[xmin, fmin, iter, ea] <- golden_ratio(y, xl, xu, tol = epsilons)

xmin

fmin

iter

ea



# use ggplot2 to create the figure
ggplot(data.frame(x = xl:xu, y = y(xl:xu)), aes(x, y)) + geom_line() + labs(x = "Length (cm)", y = "Deflection (cm)", title = "Beam Deflection") + stat_valleys(colour = "purple") + stat_valleys(y.label.fmt = "%1.2f", geom = "text", colour = "red", vjust = 1.5, aes(label = paste(..y.label.., "cm at", ..x.label.., "cm"))) + theme_bw() # Source 2
```

<br />
<br />

Using GNU Octave through RcppOctave to obtain the answer for Chapra 184 / Problem 7.16:

<br />
<br />

```{r, eval = FALSE, warning = FALSE, message = FALSE}
library(RcppOctave)

o_source(text = "
L = 600 % cm

E = 50000 % kN/cm^2

I = 30000 % cm^2

w0 = 2.5 % kN/cm

epsilons = 1 / 100 % %

xl = 0

xu = L


z = @(x) (w0 / (120 * E * I * L)) * (-x ^ 5 + 2 * L ^ 2 * x ^ 3 - L ^ 4 * x)


function [x, k] = golden_section( f, a, b, tol, maxiter, verbose )
  %---------------------------------------------------------------------------
  % Jonathan R. Senning <jonathan.senning@gordon.edu>
  % Gordon College
  % Written May 3, 1999
  % Revised May 4, 2005
  % Revised December 18, 2008 - Works with both Matlab and Octave
  %
  % Implements the golden section search as described on pages 555-556 of
  % Numerical Mathematics and Computing, 4th Edition, by Cheney & Kincaid,
  % Brooks/Cole, 1999 except that y is calculated as an offset from b rather
  % than from a.
  %---------------------------------------------------------------------------

  if ( nargin <= 5 )
    verbose = 0;
  end

  % Set things up to begin the iteration
  r  = ( sqrt( 5 ) - 1 ) / 2;   % Golden ratio
  x = a + r * ( b - a );
  y = b - r * ( b - a );
  u = f( x );
  v = f( y );

  % Main loop
  k = 0;
  if ( verbose ~= 0 )
    fprintf( '%4d: Interval is [%f, %f]; y = %f, x = %f\n', k, a, b, y, x );
  end
  while ( abs( b - a ) >= 2 * tol && k < maxiter )
    k = k + 1;
    if ( u > v )        % minimum must be in [a, x]
      b = x;
      x = y;
      u = v;
      y = b - r * ( b - a );
      v = f( y );
    else                % minimum must be in [y, b]
      a = y;
      y = x;
      v = u;
      x = a + r * ( b - a );
      u = f( x );
    end
    if ( verbose ~= 0 )
      fprintf( '%4d: Interval is [%f, %f]; y = %f, x = %f\n', k, a, b, y, x );
    end
  end

  % All done; best estimate of solution is midpoint of final interval
  x = 0.5 * ( a + b );

endfunction % needed for GNU Octave only



[xmin, iter] = golden_section(z, 0, 8, tol = 1e-08, maxiter = 100)


xmin

iter
")
```

<br />
<br />

# Example 4

"Determine the critical travel time and concentration" using the Streeter-Phelps model. (Chapra 185) / Problem 7.25

<br />
<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
import::from(pracma, fminbnd)
# import fminbnd from the pracma package


source("https://raw.githubusercontent.com/ggrothendieck/gsubfn/master/R/list.R")

o <- function(t) {
o1 <- os - ((kd * L0 / (kd + ks - ka)) * (exp(-ka * t) - exp((-kd + ks) * t))) - Sb / ka * (1 - exp(-ka * t))
return(o1)
}

os <- 10 # mg/L

kd <- 0.1 # d^-1

ka <- 0.6 # d^-1

ks <- 0.05 # d^-1

L0 <- 50 # mg/L

Sb <- 1 # mg/L/d

list[oc, tc] <- fminbnd(o, 0, 5)

oc

tc
```

<br />
<br />

Using GNU Octave through RcppOctave to obtain the answer for Chapra 184 / Problem 7.16:

<br />
<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
library(RcppOctave)

o_source(text = "
function o1 = o(t)
global os kd L0 ks ka Sb
o1 = os - ((kd * L0 / (kd + ks - ka)) * (exp(-ka * t) - exp((-kd + ks) * t))) - Sb / ka * (1 - exp(-ka * t));
endfunction % only needed for GNU Octave


global os kd L0 ks ka Sb
os = 10 % mg/L

kd = 0.1 % d^-1

ka = 0.6 % d^-1

ks = 0.05 % d^-1

L0 = 50 % mg/L

Sb = 1 % mg/L/d

[oc, tc] = fminbnd(@o, 0, 5)

oc

tc
")
```

<br />
<br />

## Sources used in the R code

Source 1<br />
anonymous function exercise in Wickham’s Advanced R book (integration of 3 functions in list) - Stack Overflow answered by jlhoward on Dec 19 2014 & answered and edited by Khashaa on Dec 19 2014. See https://stackoverflow.com/questions/27572408/anonymous-function-exercise-in-wickhams-advanced-r-book.

Source 2<br />
Using R for photobiology: R tips 3: ggpmisc adds new stats to ‘ggplot2’ by Pedro J. Aphalo, 2016-01-30. See http://www.r4photobiology.info/2016/01/ggpmisc-adds-new-stats-to-ggplot2/.

<br />
<br />

## Works Cited

Hadley Wickham, *Advanced R* (Online), “Functional programming”, http://adv-r.had.co.nz/Functional-programming.html

Jonathan R. Senning, "golden_section GNU Octave/MATLAB program, Gordon College, Revised December 18, 2008, http://www.math-cs.gordon.edu/courses/mat342/octave/golden_section.m.

Steven C. Chapra, *Applied Numerical Methods with MATLAB for Engineers and Scientists*, 2nd Edition, Boston, Massachusetts: McGraw-Hill, 2008, pages 184-185, 381.

<br />
<br />

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## Copyright and License

All R code written by Irucka Embry is distributed under the GPL-3 (or later) license, see the [GNU General Public License (GPL) page](https://gnu.org/licenses/gpl.html).

All written content originally created by Irucka Embry is copyrighted under the Creative Commons Attribution-ShareAlike 4.0 International License. All other written content remains as the copyright of the original author(s).

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