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COMPSCI1JC3-Haskell-CubicCalculator Solved

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The purpose of Assignment 1 is to write a program in Haskell that computes the solutions of a cubic equation. The requirements for Assignment 1

and for an extra credit assignment, Assignment 1 Extra Credit, are given below. You are required to do Assignment 1, but Assignment 1 Extra Credit is optional. Please submit Assignment 1 as a single Assign 1.hs file to the Assignment 1 folder on Avenue under Assessments/Assignments. If you choose to do Assignment 1 Extra Credit for extra marks, please submit it also as a single Assign 1 ExtraCredit.hs file to the Assignment 1 Extra Credit folder on Avenue in the same place. Both Assignment 1 and Assignment 1 Extra Credit are due September 29, 2019 before midnight. Assignment 1 is worth 4% of your final grade, while Assignment 1 Extra Credit is worth 2 extra percentage points.

Late submissions will not be accepted! So it is suggested that you submit a preliminary Assign 1.hs file well before the deadline so that your mark is not zero if, e.g., your computer fails at 11:50pm on September 30.

Although you are allowed to receive help from the instructional staff and other students, your submitted program must be your own work. Copying will be treated as academic dishonesty!

1      Background

Recall from high school mathematics that a quadratic equation

ax2 + bx + c = 0,

where a,b,c are real numbers with a 6= 0, has two solutions:

.

The solutions of the quadratic equation depend on the value of b2 − 4ac called the discriminant of the equation. If b2 − 4ac < 0, the two solutions are non-real complex numbers; if b2 − 4ac = 0, the two solutions are real numbers equal to each other; and if b2 − 4ac > 0, the two solutions are distinct real numbers.

Less well known is that a cubic equation

ax3 + bx2 + cx + d = 0,

where a,b,c,d are real numbers with a 6= 0, has three solutions:

where:

.

The derivation of the solutions x1, x2, and x3 is elegantly presented in the ProofWiki article “Cardano’s Formula” found at https://proofwiki.org/wiki/Cardano’s Formula.

The solutions of the cubic equation depend on the value of Q3+R2 called the discriminant of the equation as follows:

  1. If Q3 + R2 < 0, then the three solutions are distinct real numbers. However, these solutions require complex number arithmetic to compute.
  2. If Q3+R2 = 0, then the three solutions are real numbers with x2 = x3. It is possible that x1 = x2 = x3 in this case.
  3. If Q3 + R2 > 0, then x1 is a real solution, but x2 and x3 are two non-real complex solutions.

Historical note: The general solution for cubic equations was first devised by Niccol`o Fontana Tartaglia (1499–1557) in 1530 and first published by Gerolamo Cardano (1501–1576) in 1545. The general solution for quartic equations (of degree four) was discovered by Lodovico Ferrari (1522–1565) in 1540. Niels Henrik Abel (1802–1829) gave the first complete proof in 1824 that general solutions do not exist for equations of degree five or greater.

2       Assignment 1

The purpose of this assignment is to compute — using only real number arithmetic — approximations of the real number solutions of a cubic equation with floating point coefficients.

2.1     Requirements

  1. Download from Avenue Assign1 Project Template.zip which contains the Stack project files for this assignment. Modify the Assign 1.hs in the src folder so that the following requirements are satisfied.
  2. Your name, the date, and “Assignment 1” are in comments at the topof your file. macid is defined to be your MacID.
  3. The file includes a function named cubicQ of type Double -> Double -> Double -> Double that computes Q from a, b, and c. The file also includes a function named cubicR of type Double -> Double -> Double -> Double -> Double that R computes from a, b, c, and d.
  4. The file includes a function named cubicDisc of type Double -> Double -> Double that computes the discriminant from q and r.
  5. The file includes a function named cubicS of type Double -> Double -> Double that computes S from q and r. The file also includes a function named cubicT of type Double -> Double -> Double that computes T from q and r. Using the ** operator, define your own cube root function of type Double -> Double that finds cube roots of both negative and positive numbers. cubicS and cubicT should use only floating point operations and no complex number operations.
  6. The file includes a function named cubicRealSolutions of type

Double -> Double -> Double -> Double -> [Double] that computes a list of real solutions from a, b, c, and d using cubicQ, cubicR, cubicDisc, cubicS, and cubicT. The list is [] when the discriminant is negative, [x1,x2,x2] when the discriminant is zero, and is [x1] when the discriminant is positive. Here x1 and x2 are defined as above in terms of a, b, S, and T. (Note that the computation of the missing solutions require complex number arithmetic.)

  1. If the functions cubicS and cubicT are implemented properly, they should be undefined when the discriminant, Q3 +R[1], is negative since

then the value of pQ[2] + R2 is not a real number. That is, these functions should return NaN — which stands for “not a number” — when Q[3] + R2 < 0.

For example, if the cubic equation is

x3− 3x = 0,

√                              √

then by factoring it is clear that x[4] = 0, x2 = 3, and x3 = − 3. However, Q3 + R[5] = −1, so cubicRealSolutions 1 0 (-3) 0 should return [] instead of

[0.0,1.7320508075688772,-1.7320508075688772].

It is a useful exercise to compute x1, x2, and x[6] for this equation using complex number arithmetic.

  1. Your file loads successfully into GHCi and all of your functions perform correctly.
  2. The file includes two functions cubicComplexS and cubicComplexT of type Double -> Double -> Complex Double where Complex Double is the type of complex numbers whose real and imaginary parts are of type Double. You can either use a complex number type from a Haskell library or define your own. When Q3 + R2 ≥ 0, these two functions behave exactly like cubicS and cubicT except they return a real number represented as Complex Double
  3. The file includes cubicComplexSolutions of type Double -> Double -> Double -> Double -> [Complex Double] that computes a list of the three solutions from a, b, c, and d using cubicQ, cubicR, cubicDisc, cubicS, cubicT. The list is [x1,x2,x3] where x1,x2,x3 are values of type Complex Double defined as above in terms of a, b, S, and T.
  4. Your file successfully loads into GHCi and all of your functions performcorrectly.

[1] .2     Testing

You should test on your own all of the functions in the file you submit. You will be required to submit formal test plans for all subsequent assignments.

[2] Assignment 1 Extra Credit

The purpose of this extra credit assignment is to compute approximations of the complex solutions of a cubic equation with floating point coefficients. This is a very challenging assignment; do not be discouraged if you cannot complete it.

[3] .1     Requirements

[4] . Add the Extra Credit functions to the Assign 1 ExtraCredit.hs file in the src folder (not Assign 1.hs). Modify this file so that the following requirements are satisfied.

[5] . Your name, the date, and “Assignment 1 Extra Credit” are in comments at the top of your file. macid is defined to be your MacID.

[6] . The file includes the same functions cubicQ, cubicR, and cubicDisc as in the file for Assignment 1.

  • Haskell-CubicCalculator-wt54q0.zip