poly: edit multiply and add divide functions (#215)

* edit multiply function, add divison, and fix comment lines

* update readme

* fix readme

* Update README.md

poly/README.md

@@ -1,13 +1,16 @@
 # Polynomials
 
-This chapter describes functions for evaluating and solving polynomials.
-There are routines for finding real and complex roots of quadratic and
-cubic equations using analytic methods. An iterative polynomial solver
-is also available for finding the roots of general polynomials with real
-coefficients (of any order). The functions are declared in the module `vsl.poly`.
+This chapter describes functions for evaluating and solving polynomials. There are routines
+for finding real and complex roots of quadratic and cubic equations using analytic methods.
+An iterative polynomial solver is also available for finding the roots of general polynomials
+with real coefficients (of any order). The functions are declared in the module `vsl.poly`.
 
 ## Polynomial Evaluation
 
+```v ignore
+fn eval(c []f64, x f64) f64
+```
+
 The functions described here evaluate the polynomial
 
 ```console
@@ -16,20 +19,13 @@ P(x) = c[0] + c[1] x + c[2] x^2 + . . . + c[len-1] x^(len-1)
 
 using Horner's method for stability.
 
-```v ignore
-fn eval(c []f64, x f64) f64
-```
-
-This function evaluates a polynomial with real coefficients for the real variable `x`.
-
 ```v ignore
 fn eval_derivs(c []f64, x f64, lenres u64) []f64
 ```
 
-This function evaluates a polynomial and its derivatives storing the
-results in the array `res` of size `lenres`. The output array
-contains the values of `d^k P(x)/d x^k` for the specified value of
-`x` starting with `k = 0`.
+This function evaluates a polynomial and its derivatives, storing the results in the array
+`res` of size `lenres`. The output array contains the values of `d^k P(x)/d x^k` for the
+specified value of `x`, starting with `k = 0`.
 
 ## Quadratic Equations
 
@@ -43,22 +39,17 @@ This function finds the real roots of the quadratic equation,
 a x^2 + b x + c = 0
 ```
 
-The number of real roots (either zero, one or two) is returned, and
-their locations are are returned as `[ x0, x1 ]`. If no real roots
-are found then `[]` is returned. If one real root
-is found (i.e. if `a=0`) then it is are returned as `[ x0 ]`. When two
-real roots are found they are are returned as `[ x0, x1 ]` in
-ascending order. The case of coincident roots is not considered
-special. For example `(x-1)^2=0` will have two roots, which happen
-to have exactly equal values.
+The number of real roots (either zero, one or two) is returned, and their locations are
+returned as `[ x0, x1 ]`. If no real roots are found then `[]` is returned. If one real root
+is found (i.e. if `a=0`) then it is returned as `[ x0 ]`. When two real roots are found they
+are returned as `[ x0, x1 ]` in ascending order. The case of coincident roots is not considered
+special. For example `(x-1)^2=0` will have two roots, which happen to have exactly equal values.
 
-The number of roots found depends on the sign of the discriminant
-`b^2 - 4 a c`. This will be subject to rounding and cancellation
-errors when computed in double precision, and will also be subject to
-errors if the coefficients of the polynomial are inexact. These errors
-may cause a discrete change in the number of roots. However, for
-polynomials with small integer coefficients the discriminant can always
-be computed exactly.
+The number of roots found depends on the sign of the discriminant `b^2 - 4 a c`. This will
+be subject to rounding and cancellation errors when computed in double precision, and will
+also be subject to errors if the coefficients of the polynomial are inexact. These errors may
+cause a discrete change in the number of roots. However, for polynomials with small integer
+coefficients the discriminant can always be computed exactly.
 
 ## Cubic Equations
 
@@ -72,13 +63,93 @@ This function finds the real roots of the cubic equation,
 x^3 + a x^2 + b x + c = 0
 ```
 
-with a leading coefficient of unity. The number of real roots (either
-one or three) is returned, and their locations are returned as `[ x0, x1, x2 ]`.
-If one real root is found then only `[ x0 ]`
-is returned. When three real roots are found they are returned as
-`[ x0, x1, x2 ]` in ascending order. The case of
-coincident roots is not considered special. For example, the equation
-`(x-1)^3=0` will have three roots with exactly equal values. As
-in the quadratic case, finite precision may cause equal or
-closely-spaced real roots to move off the real axis into the complex
-plane, leading to a discrete change in the number of real roots.
+with a leading coefficient of unity. The number of real roots (either one or three) is
+returned, and their locations are returned as `[ x0, x1, x2 ]`. If one real root is found
+then only `[ x0 ]` is returned. When three real roots are found they are returned as
+`[ x0, x1, x2 ]` in ascending order. The case of coincident roots is not considered special.
+For example, the equation `(x-1)^3=0` will have three roots with exactly equal values. As
+in the quadratic case, finite precision may cause equal or closely-spaced real roots to move
+off the real axis into the complex plane, leading to a discrete change in the number of real roots.
+
+## Companion Matrix
+
+```v ignore
+fn companion_matrix(a []f64) [][]f64
+```
+
+Creates a companion matrix for the polynomial
+
+```console
+P(x) = a_n * x^n + a_{n-1} * x^{n-1} + ... + a_1 * x + a_0
+```
+
+The companion matrix `C` is defined as:
+
+```
+[0 0 0 ... 0 -a_0/a_n]
+[1 0 0 ... 0 -a_1/a_n]
+[0 1 0 ... 0 -a_2/a_n]
+[. . . ... . ........]
+[0 0 0 ... 1 -a_{n-1}/a_n]
+```
+
+## Balanced Companion Matrix
+
+```v ignore
+fn balance_companion_matrix(cm [][]f64) [][]f64
+```
+
+Balances a companion matrix `C` to improve numerical stability. It uses an iterative scaling
+process to make the row and column norms as close to each other as possible. The output is
+a balanced matrix `B` such that `D^(-1)CD = B`, where `D` is a diagonal matrix.
+
+## Polynomial Operations
+
+```v ignore
+fn add(a []f64, b []f64) []f64
+```
+
+Adds two polynomials:
+
+```console
+(a_n * x^n + ... + a_0) + (b_m * x^m + ... + b_0)
+```
+
+Returns the result as `[a_0 + b_0, a_1 + b_1, ..., a_k + b_k ...]`.
+
+```v ignore
+fn subtract(a []f64, b []f64) []f64
+```
+
+Subtracts two polynomials:
+
+```console
+(a_n * x^n + ... + a_0) - (b_m * x^m + ... + b_0)
+```
+
+Returns the result as `[a_0 - b_0, a_1 - b_1, ..., a_k - b_k, ...]`.
+
+```v ignore
+fn multiply(a []f64, b []f64) []f64
+```
+
+Multiplies two polynomials:
+
+```console
+(a_n * x^n + ... + a_0) * (b_m * x^m + ... + b_0)
+```
+
+Returns the result as `[c_0, c_1, ..., c_{n+m}]` where `c_k = ∑_{i+j=k} a_i * b_j`.
+
+```v ignore
+fn divide(a []f64, b []f64) ([]f64, []f64)
+```
+
+Divides two polynomials:
+
+```console
+(a_n * x^n + ... + a_0) / (b_m * x^m + ... + b_0)
+```
+
+Uses polynomial long division algorithm. Returns `(q, r)` where `q` is the quotient and `r`
+is the remainder such that `a(x) = b(x) * q(x) + r(x)` and `degree(r) < degree(b)`.