THE SOLUTION OF THE CUBIC EQUATIONS

Hairetdinov R. F.

THE SOLUTION OF THE CUBIC EQUATIONS.

2005 year

Introduction.

The studying of the solution of the cubic equations usually concerns to a course of the higher algebra and connects with the theory of functions of the complex variable (TFCV). Thus formula Cardano is used which is applicable only in special cases at the certain correlations of factors of the cubic equation. In other cases final formulas for calculation of roots of the cubic equations are difficult and accessible only after studying TFCV.

In this article the way of the solution of the cubic equations is expounded with the help of quickly converging recurrent formulas which are received by using of the universal cubic equations. Thus there is no necessity of studying TFCV and it is enough to acquire knowledge in the field of mathematics received in the secondary school.

These offered recurrent formulas for the solution of the cubic equations are rather easy and allow to get solutions with accuracy and for a short period of time. Thus we can use both usual pocket scientific calculators, and programmable, that is especially convenient for using of recurrent formulas.

The studying and the development of this offered way of the solution of the cubic equations allow to master the bases of the solution of the degree equations, to receive skills of using of recurrent formulas and programming at calculation with the help of recurrent formulas even at a secondary school.

1. The using of formula Cardano

The cubic equation in a general view is known (L1)
(1)

that it is possible to transform to the reduced kind
(2)

having substituted in (1) instead of х expression
(3)

For all that

(4)

(5)

The cubic equation of kind (2) we name the reduced cubic equation (for brevity: RCE).

Further for the simplification of writing of formulas RCE we drop an index «y» at factors В_y and D_y, and the formula RCE we write down as

(6)

At the certain (mentioned below) correlation between factors B and D the value of a material root y₁ RCE can be calculated with the help of formula Cardano that is below in the modified kind:

(7)

where - factor RCE. (8)

From the formula (7) it is clear, that for the receipt of material numbers under the sign of a square root there should be a number, the greater or equal to zero, i.e.

In cases whent_y > 6,75, under the formula (7) the imaginary (complex) numbers turn out, and for the receipt of values of roots RCE as material numbers it is necessary to use the theory of functions of complex variable ( TFCV) (L1).

In section 3 and further the way of solution RCE is offered which is different from the way of Cardano and it does not demand the using of TFCV for calculation of material values of roots RCE at any sizes of a factor t_y.

2. The calculation of the second and third roots of the cubic equation.

For the explanation of some moments we consider RCE with the concrete numerical data:

The equations above have 3 various material roots and in the qualityy₁ (the first root) choose that root, which has a maximal (on the module) size, and in the qualityy₂ and y₃ are the other two roots of the same sign.

(1)

In the examples with the numerical data y₁ = + 10 and – 10; y₂ = – 2 and + 2;
y₃ = – 8 and + 8, d = + 16.

The size d (equal to the product y₂ ∙ y₃ with identical signs) always has a sign «+». The signsy₁ and D are always different. There are examples of RCE below with one material root and two imaginary roots.

In the examples above y₁ = +10 and –10 — are material roots, y₂ and y₃ – are the imaginary roots which are turning out after the solution of quadratic equations of decomposition RCE.

The quadratic equation of the decomposition (1) looks like

. (2)

The formula (2) can be used at calculation of sizes of roots y₂ and y₃

(3)

(4)

When t_y < 6,75 and y_2,3 – are imaginary roots, also there is no necessity to calculate numerical parameters of imaginary roots then the decomposition (1) can be presented as the solution RCE, and it is possible to check of correctness of solution RCE by checking the equality received from formulas of the decomposition (1):

(5)

Note1. At the reference to formulas of the other unit we designate them as (N – K), where N is a number of the section, K is a number of the formula in this section.

Note2. After the definition of roots RCE the calculation of roots of the initial solved equation (1 – 1) is under the formula (1 – 3)

3. The universal cubic equation of kind 1.

Let (1)

After the substitution (1) in the formula (1 – 6) and the transformations we receive:

(2)

where (3)

The formula (2) we name it as the formula of the universal cubic equation of
a kind 1 (for brevity further UCE – 1).

Under the formula (3) the factor RCE is calculated at the solution by UCE – 1. Under the formula (1) the first root RCE on the size "k" (the first root UCE – 1) and the size of the factor D in the formula (1 – 6) is calculated. We take the size k in formulas (1) and (2) from zero up to (+ ∞), i.e. k ≥ 0.

The size k_y in formulas (2) and (3) can accept the value from (-∞) up to (+ ∞).

The universality of the formula (2) is that any cubic equation of a kind (1 – 1) can be reduced unequivocally to the formula (2), the solution of the equation (2) depends only on one parameter the factor k_y, the dependences k_y = f(k) и
k = φ(k_y) under the condition of k > 0 are unequivocal at any sizes k_y (от – ∞ до
+ ∞).

The cubic function under the formula (2) can be represented at k > 0 as the graph k_y = f (k) or to present as the detailed tables.

The algorithm of solution RCE through UCE – 1 consists of the calculation of the parameterk_y under the formula (3), the definition the of size «k» on the size of the parameter (factor) k_y with the formula (2) and the calculation of the size of the first root y₁ RCE under the formula (1). For the solution UCE – 1, except the graph
k_y = f (k) or tables, it is possible to use the formulas received further according the text. In this article the recurrent formulas are used on which calculations are conducted by a method of iteration cyclically before the reception of the result with necessary exactness. In each new iteration, the result of the previous iteration is used.

The recurrent formula is suitable for using if the mistake of result, after each iteration decreases, it means, that the recurrent formula possesses convergence.

As a rule, the recurrent formulas possess convergence not at all possible values of a parameter (the factor of the cubic equation), but only in some working area of a parameter with concrete borders for each recurrent formula.

Also the recurrent formulas differ from each other in the speed of convergence, i.e. in the size of the ratio of the module of a result’s mistake received in the previous iteration, to the module of a result’s mistake received in the subsequent iteration.

If the speed of convergence is more than a unit the formula has convergence at the given value of a parameter.

From some set of the recurrent formulas which can be received from the formula (2), the following have been selected on the speed of convergence:

(4)

(5)

The method of the calculation of the speed of convergence of the recurrent formulas we show by the example of the formula (4).

At k = 2 under the formula (2) k_y = 3,5 .

Let suppose that the result of the previous calculation was k_i = 1,99 with a mistake (on the module) |∆₁| = k – k_i = 2 – 1,99 = 0,01 .

Under the formula (4) we receive the size k_i+1 = k₂

The module of a mistake of the result, received after (i + 1) cycle of calculations is |∆₂| = 2,000628042 – 2,0 = 0,000628042 .

The speed of convergence in the set conditions is

Let's calculate the speeds of the convergence which are turning out at a variation of set conditions.

Let k₁ = 2,01, |∆₁| = 2,01 – 2,0 = 0,01

Let k₁ = 3,0, |∆₁| = 3 – 2 = 1

Let k₁ = 1,0, |∆₁| = 2 – 1 = 1

From the comparison of results of calculations it is possible to conclude, that the speed of a convergence depends on the size and the sign of a mistake of the initial value of the unknown quantity, and for the comparison of various formulas on the speed of convergence it is necessary to set identical initial conditions (an identical sign and the size of a mistake of the first value of variable). At small sizes of a set mistake |∆₁| the sizes of speed of convergence at a variation |∆₁| differ a little on the size from each other. At the big sizes of a set mistake the speed of convergence can differ from the speeds received at small sizes of a mistake |∆₁|, and it should be meant at the further calculations.

Then the technique of reception of recurrent formulas for the solution UCE in focal points is.

Let suppose that k = k₀ + ∆k, where k₀ is a focal point which can be equaled to any chosen value from zero up to (+ ∞); ∆k is a deviation from this focal point.

Let's substitute this expression in the formula (2) and we solve the received equation relatively ∆k, considering that k₀ is a constant quantity.

(6)

The formula (6) represents the general recurrent formula for the solution UCE – 1 in a focal point k₀.

At k₀ = 0 from the formula (6) we receive:

(7)

From the second variant of the formula (7) we receive:

(8)

Before the sign √ in formulas (6), (7), (8) the sign «+» is taken after checking of formulas in calculations.

Atk₀ = 1 from the formula (6) we receive

(9)

For the orientation we calculate values k_y and k for RCE with double roots

Under the formula (3)

From the formula (1) .

By the technique which was given before we check up formulas (4), (5), (7), (8), (9) on the convergence and the speed of convergence. The results of calculations (characteristic points) are in Table 1.

The analysis of Table 1 allows to draw the following conclusions.

The formula (4) possesses convergence at k_y > –0,61 , the speed of convergence of the formula accrues with increase k_y.

The formula (5) possesses convergence at k_y > –0,938 , the speed of convergence of the formula is maximal in a narrow range at k_y ≈ 0.

Formulas (7), (8), (9) possess convergence at all possible values k_y, but each formula possesses high speeds of convergence in different areas of values k_y, the formula (7) at k_y << - 1, the formula (8) at k_y >> 1, the formula (9) at k_y ≈ 0.

Table 1

k	k_y	k₁	The speeds of convergence of the formula №
k	k_y	k₁	4	5	7	8	9	10	11	12	13
0,01	-99,99	0,011	─	─	10⁶	2,04	112,3			10⁸	4∙10¹⁴
0,1	-9,99	0,11	─	─	10³	2,05	11,3			10⁴	4∙10⁷
0,2	-4,96	0,21	─	─	126,0	2,04	6,9			869,4	4∙10⁵
0,3	-3,24	0,31	─	─	38,1	2,07	7,7			293,3	4∙10⁴
0,4	-2,34	0,41	─	─	16,6	2,11	6,7			112,7	8∙10³
0,5	-1,75	0,51	─	─	9,0	2,26	8,9			80,4	4∙10³
0,6	-1,306	0,61	─	─	5,6	2,44	13,8			77,2	2,5∙10³
0,7	-0,938	0,71	─	1,58	3,94	2,7	35,6			140,2	4∙10³
0,8	-0,61	0,81	1,01	3,16	2,97	3,0	62,1		995,6	184,4	4,9∙10³
0,9	-0,3011	0,91	1,46	8,08	2,38	3,44	271,9		2∙10⁵	647,1	1,8∙10⁴
1,0	0	1,01	2,01	∞	2,01	4,01	∞		∞	∞	∞
1,1	0,3009	1,11	2,64	12,05	1,76	4,64	332,0		3∙10⁶	584,3	2,2∙10⁴
1,2	0,6066	1,21	3,48	7,12	1,59	5,47	91,9	1,7∙10³	1,3∙10⁵	148,1	6∙10³
1,3	0,9207	1,31	4,36	5,51	1,46	6,37	45,16	6,8∙10³	2,75∙10⁴	65,9	3,9∙10³
1,36	1,114	1,35	4,99	4,97	1,38	7,01	33,1	1,5∙10⁴	1,5∙10⁴	45,6	3∙10³
1,5	1,583	1,51	6,79	4,26	1,3	8,76	19,6	9,7∙10⁴	5,9∙10³		2,5∙10³
		1,597	8,047	4,006				2,7∙10⁵	4,1∙10³
2	3,5	2,01	15,9	3,42	1,11	17,9	7,51	10⁷	1,6∙10³		2,9∙10³
3	8,666	3,01	53,8	3,11	1,03	55,9	3,58	2∙10¹⁰	904,8		10⁴
10,0	99,9	10,1	2019	3,01	1,01	2002	1,98	6∙10¹⁹	743,7		6∙10⁶

It is necessary to note, that in the field of the convergence it is possible to make cascade formulas of recurrent formulas, i.e. to insert one or several formulas into one another, and the speeds of convergence of cascade formulas are roughly equal to the product of speeds of each formula included in it at identical values of a parameter (k_y).

For the calculations in area (three material roots) from formulas (4) and (8) it is possible to construct multicascade formulas, but from the formula (4) multicascade formulas are more simple on structure.

Let suppose that , then the six-cascade formula on the basis of the formula (4) looks so:

(10)

The formula (10) is applicable thanks to the high speeds of convergence at k_y > 0,6. Formulas (5) and (9) also allow to build cascade formulas with the high speeds at k_y ≈ 0, but the formula (5) is more preferable, for example, the six-cascade formula is below submitted on the basis of the formula (5)

(11)

In a range k_y < 0 the application of cascade formulas is possible on the basis of formulas (7) and (9).

Let's substitute in (9) formula (7), we receive:

(12)

Taking into account the convergence of formulas (7), (8), (9) at all possible values k_y, it is possible to unite these formulas in various combinations and to receive universal recurrent formulas for the solution UCE – 1. The formula is below constructed on a pattern (7) ∙ (8) ∙ (9).

(13)

The speeds of convergence of formulas (10), (11), (12), (13) are in Table 1. As the first (initial) value k_i = k₁ in recurrent formulas UCE – 1 it is possible to substitute number k₁ = 1 or the number which has been picked up on the value k_y from
Table 1.

4. Formula UCE–2.

Let (1)

After the substitution (1) in formula RCE (1 – 6) and transformations we receive the formula of the universal cubic equation of kind 2 (for brevity: UCE-2)

(2)

where . (3)

At B < 0 the equation (2) becomes:

(4)

m = 1 ÷ (+ ∞), m_y = 0 ÷ (+ ∞).

At B > 0 we have:

(5)

m = 0 ÷ (- ∞), m_y = 0 ÷ (- ∞).

Let's calculate the sizes m_y and m for RCE with double roots, and also for RCE, at which maximal and minimal (on the module) sizes of roots differ, for example, in 1000 times.

Under formula (3)

From formula (1) (6)

From the examples it is visible, that the area of valuesm_y, at which all 3 roots RCE have material sizes, is between 0 and and the size m is in area with borders 1 and

The following most productive recurrent formulas have been received and selected from the formula (4), also after using of the substitution:

(7)

(8)

(9)

The results of calculations for the speed of convergence of the received recurrent formulas with the use of a technique of unit (3) are collected in Table 2.

Table 2.

m	m_y	m₁	The speeds of convergence of the formula №
m	m_y	m₁	7	8	9	10	11
1	0	1,01	3,0	∞	∞	729	∞
1,05	0,1076	1,051	3,3	21,5	922,0	1291	7,8∙10⁸
1,1	0,231	1,11	3,63	11,6	262,1	2,2∙10³	1,8∙10⁷
1,15	0,3708	1,16	3,97	8,26	131,0	3,9∙10³	2,2∙10⁶
1,1547	0,384900179	1,1447			124,23		1,917∙10⁶
1,155	0,38579	1,154	4,001	7,981	123,9	4,1∙10³	1,9∙10⁶
1,2	0,528	1,21	4,32	6,59	82,0	6,4∙10³
1,5	1,875	1,51	6,75	3,62	22,0	9,4∙10⁴
2	6,0	2,01	12,0	2,67	10,0	2,98∙10⁶
3	24,0	3,01	27,0	2,25	5,5	3,8∙10⁸
10	990	10,1	300,0	2,03	2,7	7∙10¹⁴
100	999900	100,1	3∙10⁴	2,0	2,06	7∙10²⁶

After the analysis of Table 2 it is possible to draw the following conclusions.

All three formulas possess convergence at m_y > 0.

The formula (7) increases the speed of convergence at increase m_y. Formulas (8) and (9) possess a maximum of the speed at m_y → 0. The formula (9) possesses at small m_y considerably the big speeds, than the formula (8).

All three formulas allow to build multicascade formulas. For example, we receive the six-cascade formula on the basis of the formula (7):

(10)

On the basis of the formula (9) it is possible to construct two, three, four-cascade formulas with the speeds of convergence at m_y = 0,38579.

V₂ = 1,5∙10⁴; V₃ = 1,9∙10⁶; V₄ = 2,3∙10⁸

The three-cascade formula is below resulted on the basis of the formula (9):

(11)

The speeds of convergence of formulas (10), (11) are resulted in Table 2. The first (initial) value m_i = m₁ для формул (7) ÷ (11) is possible to take number
m₁ = 1.

For UCE – 2 at B > 0, m_y < 0 from the formula (5) it is possible to receive the following most productive recurrent formulas:

(12)

(13)

Having substituted in the formula (5) expressions |m| = 1 + |∆m| и |m| = 0,5 + |∆m|, we receive:

(14)

(15)

The speed of convergence of formulas (12), (13), (14), (15) are in Table 3.

After the analysis of Table 3 it is possible to draw the following conclusions.

The formula (12) possesses convergence at m_y < - 0,816, the speed of convergence of the formula grows with increase in the module |m_y|.

The formula (13) possesses convergence at m_y < 0, a maximum of speed at |m_y|→0.

The formula (14) possesses a maximum of speed at m_y = –2,0 , and on both sides from the value m_y = –2,0 speeds fall.

The formula (15) possesses a maximum of speed at m_y = –0,625, and on both sides from the value m_y = –0,625 speeds fall.

Table 3.

- m	- m_y	- m₁	12	13	14	15	16	17
0,1	0,101	0,11	─	102,1	─	5,37		10⁴
0,2	0,208	0,21	─	27,09	─	11,3		733,8
0,3	0,327	0,31	─	13,17	1,53	30,6		173,4
0,4	0,464	0,41	─	8,29	3,05	146,7		68,7
0,5	0,625	0,51	─	6,03	5,94	∞		36,3
0,6	0,816	0,61	1,06	4,8	11,9	207,1		23,0
0,7	1,043	0,71	1,45	4,06	26,3	60,8		16,4
0,8	1,312	0,81	1,9	3,58	71,8	31,5		12,8
0,9	1,629	0,91	2,4	3,25	341,7	20,98		10,5
1,0	2,0	1,01	2,98	3,01	∞	15,03	78,8	9,06
1,1	2,431	1,11	3,62		462,1		171,7
1,2	2,928	1,21	4,31	2,7	132,0		345,0
1,3	3,497	1,31	5,06		66,5		655,5
1,5	4,875	1,51	6,74	2,45	30,0		2∙10³
2,0	10	2,01	11,99	2,25	12,01		2∙10⁴
10,0	1010	10,1	29,9	2,02	2,73		8∙10⁵

On the basis of the formula (12) it is possible to build multicascade formulas with high speeds of convergence. For example the four-cascade formula (16) is below.

(16)

On the basis of the formula (13) it is also possible to construct two, three, four -cascade formulas. For example the two-cascade formula (17) is below.

(17)

The speeds of convergence of formulas (16), (17) are in Table 3. The initial values m_i = m₁ for formulas (12) ÷ (17) can be taken m₁ = –1.

The fields of application of formulas can be chosen according Table 3 depending on the speed of convergence of the concrete formula.

5. Formulas UCE– 3 and UCE – n.

Let (1)

Having substituted (1) in formula RCE (1 – 6), we receive formula UCE – 3:

(2)

where (3)

If we use the formula

(4)

we receive formula UCE– n:

(5)

Between formulas (2) and (5) there is a connection as

(6)

Accordingly, formulas (1) and (4) can be written down as:

(7)

It is possible later on to transform all formulas received for the solution UCE with use of formulas (6).

Let's calculate sizes t_y, t and n for RCE with double roots:

Under formula (3)

From formula (1)

From formula (4)

From formulas (2) and (5) it is possible to receive the following most effective recurrent formulas:

(8)

(9)

(10)

(11)

Having made substitutions t = –1 + ∆t and t = –0,5 + ∆t in the formula (2), after transformations we receive recurrent formulas for solution UCE – 3 in the vicinity of points t₀ = - 1 and t₀ = –0,5.

(12)

(13)

In table 4 the results of calculations are shown for the speed of convergence by a technique of unit 3 of recurrent formulas (8), (12), (13).

Table 4.

t	t_y	t₁	The speeds of convergence of the formula №
t	t_y	t₁	8	12	13	14	15
-0,9	-7,29	-0,91	─	970,7	59,7	9∙10⁵
-0,8	-2,56	-0,81	─	110,8	48,8	10⁴
-0,75	-1,687	-0,76	─	52,9	53,0	2,7∙10³
-0,7	-1,143	-0,71	─	27,9	64,4	781,2
-0,6	-0,54	-0,61	1,98	9,05	161,1	82,0	15,4
-0,55	-0,369	-0,56	2,43	5,238	510,27	27,4	35,2
-0,5	-0,25	-0,51	2,97	2,92	∞	8,5	78,7
-0,4	-0,106	-0,41	4,47	─	57,4		400,9
-0,3	-0,038	-0,31	6,96	─	6,56		2,3∙10³
-0,2	-0,01	-0,21	11,94	─	─		2∙10⁴
-0,1	-0,00111	-0,11	26,8	─	─		5∙10⁵
0	0	0,01	∞				∞

The continuation of Table 4.

t	t_y	t₁	8	9	11	16	15	18	17
0,1	0,000909	0,11	33,09	─	─		10⁶
0,2	0,0066	0,21	18,04	2,49	─		10⁵
0,5	0,0833	0,51	9,01	3,049	─	86,4	0,6∙10³		803,4
1,0	0,5	1,01	6,0	4,034	4,0	264,8	1,3∙10³		4,3∙10³
2,0	2,666	2,01	4,5	6,027	27	1319,4	411,8	729,0	4,7∙10⁴
3,0	6,75	3,01	4,0	8,02	80,1	4∙10³	256,0	6,4∙10³	2,6∙10⁵
5,0	20,88	5,01	3,6	12,0	324,5	2∙10⁴	167,0	10⁵	2,9∙10⁶
10,0	90,909	10,1	3,31	22,2	2319,8	2,4∙10⁵	120,0	5∙10⁶	1,1∙10⁸

After the analysis of Table 4 it is possible to draw the following conclusions.

The formula (8) possesses a maximum of speed of convergence at t_y = 0, the working area of values of a parameter t_y ≥ –0,25.

The formula (9) possesses convergence at t_y > 0, with increase in the size t_y the speed grows.

The formula (11) possesses convergence at t_y ≥ 0,5, the speed of convergence of the formula quickly increases with growth t_y.

The formula (12) possesses convergence at t_y ≤ –0,25, the speed of the formula quickly increases at t_y → – ∞.

The formula (13) possesses a maximum of speed at t_y = –0,25, the working area of a parameter t_y from (– ∞) to(– 0,1066).

On the basis of formulas (8), (9), (11), (12) it is possible to receive cascade formulas.

(14)

(15)

From formula (9) we get

(16)

(17)

(18)

The speeds of convergence of formulas (14) ÷ (18) are shown above in Table 4. The working values t_y and rough-approached values t_i = t₁ for the received formulas are possible to take from Table 4 proceeding from the value t_y and sizes of the speed of convergence of the given formula.

5.1 Formula UCE – n+.

At n < 1, t_y < 0 the equation (5 – 5) is more convenient for solving at the replacement of a variable «n» on other variable

or (1)

After the substitution (1) in (5 – 5) we receive the formula UCE – n+:

. (2)

After the transformations from the formula (2) it is possible to receive the following most effective recurrent formulas:

(3)

(4)

After the substitution U = 2 + ∆U in the formula (2) it is possible to receive:

(5)

The results of calculations for the speed of convergence of formulas (3), (4), (5) are in Table 5 by a technique of unit 3.

After the analysis of Table 5 it is possible to draw the following conclusions.

The formula (3) possesses the high speed of convergence at t_y → 0 , the speed decreases with reduction t_y (increase in the module |t_y|).

The formula (4) possesses the speed of convergence more than 3 at t_y ≤ –0,25 , and if the module |t_y| is more, then the speed of convergence is more too.

The formula (5) possesses a maximum of speed at t_y ≈ –0,25 , and on both sides from this point the speeds fall.

Table 5.

U	t_y	U₁	The speeds of convergence of the formula №
U	t_y	U₁	3	5	4	6	7	8
1	0	1,01	∞	─	─	∞
1,1	-0,00082	1,11	30,19	─	─	8,3∙10⁵
1,3	-0,0159	1,31	10,05	─	─	10⁴
1,5	-0,0555	1,51	6,033	─	─	1,3∙10³
1,7	-0,1186	1,71	4,308	10,75	─	344,4	115,5
1,75	-0,1377	1,76	4,02	18,18	─	261,4	330,6
1,8	-0,158	1,81	3,769	32,6	─	200,5	1066,4
1,9	-0,2019	1,91	3,35	165,04	─	125,9	2,7∙10⁴
2,0	-0,25	2,01	3,014	∞	3,001	82,5	∞
2,1	-0,3018	2,11	2,74	235,4	3,968		5,5∙10⁴
2,3	-0,415	2,31	─	34,16	6,285		1,1∙10³
2,5	-0,54	2,51	─	15,23	9,154		231,9	83,7
2,6	-0,6059	2,61	─	11,6	10,81		134,7	116,8
2,7	-0,6739	2,71	─	9,288	12,62		86,2	159,3
3,0	-0,8888	3,01	─	─	19,058			383,3
5,0	-2,56	5,01	─	─	111,2			1,2∙10⁴
10,0	-7,29	10,1	─	─	980,6			9,6∙10⁵

On the basis of formulas (3), (5), (4) it is possible to construct cascade formulas with the increased speeds of convergence.

(6)

(7)

(8)

The speeds of convergence of formulas (6), (7), (8) are shown in Table 5. It is possible to take the initial values U_i = U₁ for formulas (3) and (6) U₁ = 1, for formulas (5) and (7) U₁ = 2, for formulas (4) and (8) U₁ = 3, or the proper value t_y according Table 5.

5.2 The formula UCE – 3 +.

Let's replace in the formula (5 – 2) a variable t on the other variable d with the substitution (1)

Let's designate then (2)

(3)

The received formula (3) we name the formula UCE – 3 +.

At the change t in a range from 0 up to – 1 the size d varies from + ∞ up to + 1, the size d_y – from + ∞ to 0.

From the formula (3) with the using of substitutions

we receive the formulas for the solution UCE – 3 + in a vicinity of points d₀ = 1 и d₀ = 2.

(4)

(5)

From the formula (5 – 8) with the use of formulas (1) and (2) we receive

. (6)

The speeds of convergence of the formulas (4), (5), (6) designed by a technique of Unit 3. They are in Table 6.

The analysis of Table 6 allows to draw the following conclusions.

The formula (4) possesses a maximum of the speed of convergence at d_y → 0, and at increase d_y speed falls.

The formula (5) possesses a maximum of the speed at d_y ≈ 4 and on both sides from a point d_y =4 speeds fall.

The formula (6) possesses convergence at d_y > 0,784 and the speed grows with increase d_y.

Table 6.

d	d_y	d₁	4	5	6	7	8	9
1	0	1,01	∞	─	─	∞
1,1	0,121	1,11	142,1	─	─	2,8∙10⁶
1,3	0,507	1,31	26,5	─	─	1,8∙10⁴
1,4	0,784	1,41	18,2	7,5	1,22	6∙10³
1,5	1,125	1,51	14,0	13,9	1,5	2,7∙10³	194,3
1,6	1,536	1,61	11,4	26,9	1,82	1,4∙10³	725,7
1,8	2,592	1,81	8,58	51,8	2,4	631,6	2,3∙10⁴
1,9	3,249	1,91	7,69	701,7	2,7	454,7	4,9∙10⁵
2,0	4,0	2,01	7,01	∞	3,02	344,4	∞	83,7
2,1	4,851	2,11	6,47	902,1	3,32	270,8	8∙10⁵	122,2
2,3	6,87	2,31	5,68	124,2	3,92	183,2	1,5∙10⁴	237,3
2,5	9,375	2,51	5,12	54,0	4,52		2,9∙10³	419,5
2,7	12,3	2,71		32,6	5,12		10³	690,5
2,8	14,1	2,81		27,0	5,42		730,1	867,0
3,0	18,0	3,01		20,0	6,02		400	1319,1
5,0	100,0	5,01			12,0			2∙10⁴
10,0	900	10,1			27,2			5,5∙10⁵

On the basis of formulas (4), (5), (6) it is possible to build cascade formulas with the increased speeds of convergence.

(7)

(8)

(9)

The speeds of convergence of formulas (7), (8), (9) are collected in Table 6. As the first initial value d_i = d₁ in formulas (4) ÷ (9) it is possible to substitute number
d₁ = 2.

6. The transformation of formulas UCE.

As all UCE are derivatives from RCE it is possible to transform formulas of the solution of one kind of UCE to the formulas of the solution of other kinds RCE.

First of all the formulas with the big speeds of convergence are interesting for the solution UCE in the area where there are three material roots (i.e. areas t_y ≥ 6,75).

At the comparison of such formulas the formulas (4 – 9) and (5 – 11) for the solution UCE – 2 and UCE – 3 are suitable.

Taking into account, that in the field of three material roots there are factors k_y, m_y and t_y > 0, also there are variables k, m, t, which are more than 0 we establish connections between the factors and the variables of the different UCE in these conditions.

(1)

At m_y > 0 B < 0, therefore

(2)

(3)

Equating to each other mentioned formulas above, we receive:

(4)

(5)

(6)

Let's write out the formula (4 – 9) for the solution UCE – 2

Let's transform the formula (4 – 9), having substituted in it formulas (4) and (3):

(7)

Let's transform the formula (4 – 9), having substituted in it formulas (2) and (6)

(8)

Let's write out the formula (5 – 11) for the solution UCE – 3

Let's transform the formula (5 – 11), having substituted in it formulas (1) and (5)

(9)

On the basis of formulas (7), (9), (8) we construct three-cascade recurrent formulas with the increased speeds of convergence:

(10)

(11)

(12)

Let's consider, how the size of a mistake of result varies at the substitution of the different values t_i = t₁ in formulas (5 – 9) and (5 – 11).

For example we take the value t = 3 and t_y = 6,75.

If t_i = t₁ = ∞, then (13)

, error

If , then (14)

At t₁ = ∞

At t₁ = 1

At (15)

The calculations under the formula (8) at give us:

(16)

The similar calculations for UCE – 1 and UCE – 2 give the following formulas:

From formula (3 – 4) (17)

From formula (7) (18)

From formula (4 – 9) for УКУ – 2 at m = 1

(19)

Contents

Introduction ……………………………………………………………...	1
1. The using of formula Cardano …………………………………………...	2
2. The calculation of the second and third roots of the cubic equation …….	3
3. The universal cubic equation of kind 1 …………………………………..	4
4. Formula UCE – 2 ………………………………………………………...	11
5. Formulas UCE – 3 and UCE – n ………………………………………...	16
5.1. Formula UCE – n + ……………………………………………………..	19
5.2. The formula UCE – 3 + ………………………………………………...	21
6. The transformation of formulas UCE…………………….........................	24

Hairetdinov Robert Farrahovich

THE SOLUTION OF THE CUBIC EQUATIONS.

Literature:

1. I.N.Bronstein, K.A.Semendjaev, The Reference book on Mathematics, edition 5, Moscow, 1955, 608 pages.

2. R.F.Hairetdinov, The Solution of the Cubic Equations, edition 2, Ufa, Russia, 2005, 50 pages.

Translator Sharafutdinova Elena Sergeevna.

mailto:yravneniya@narod.ru

Hairetdinov R. F.