Square root of a multivector in 3D Clifford algebras

. The problem of square root of multivector (MV) in real 3D ( n = 3 ) Clifford algebras Cl 3 , 0 , Cl 2 , 1 , Cl 1 , 2 and Cl 0 , 3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (inﬁnitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved.


Introduction
The square root and the power of a real number are the two simplest nonlinear operations with a long history [10]. Solution by radicals of a cubic equation was first published after very long period, in 1545 by G. Cardano. Simultaneously, a concept of square root of a negative number as well as of complex number have been developed [10]. A. Cayley was the first to carry over the notion of the square root to matrices [6]. In the recent book by Higham [11], where an extensive literature is presented on nonlinear functions of matrices, two sections are devoted to analysis of square and pth roots of general matrix. The existence of matrix square root here is related to matrix positive eigenvalues. In the context of noncommutative Clifford algebra (CA), the main attempts up till now were concentrated on the square roots of quaternions [19,20] or their derivatives such as coquaternions (also called split quaternions) or nectarines [9,20,21]. The square root of biquaternion (complex quaternion) was considered in [22]. All they are isomorphic to 2D Clifford algebras (CA) Cl 0,2 , Cl 1,1 , Cl 2,0 , and therefore, the quaternionic square root analysis can be easily carried over to CA. In this paper, we shall be interested in higher 3D CAs, where the main object is the 8-component multivector (MV).
The understanding and investigation of square roots in noncommutative CAs is still in its infancy. The published formulas are not complete or even erroneous in case of a general MV [7]. The most akin to present results are the investigations of conditions for existence of square root of −1 [12,13,22] in the Clifford-Fourier transforms and CAbased wavelet theory [14]. In contrast to complex number Fourier transform (we shall remind that complex number algebra is isomorphic to Cl 0,1 ), the existence of variety of square roots of −1 in noncommutative CAs allows to create new geometric kernels for Fourier transforms and wavelets as well as for left-right and double-sided Fourier transforms. More important is the fact that the CA allows to extend analysis beyond complex numbers [5,13]. In paper [7], along with functions for MVs, the authors provide several formulas for square roots in Cl 3,0 . However, the general formula in [7] as shown in this article is incorrect.
Since the square root is a nonlinear operation, a nonstandard approach is needed. For this purpose, we have applied the Gröbner-basis algorithm to analyze the system of polynomial equations that ensue from MV equation A 2 = B, where A and B are the MVs. The Gröbner basis is accessible in symbolic mathematical packages such as Mathematica and Maple. In this paper, the formulas for practical calculations of the square root of MVs in all 3D Clifford algebra are presented. In case of MV, we have found that a characteristic property of roots in 3D Clifford algebras is that the MV may have no roots, a single or multiple isolated roots, or even an infinitely many roots in the parameter space.
In Section 2, the needed notation is introduced. In Section 3, the method of calculation of roots in described. Then, we prove that in 3D CAs, the square root of MV can be expressed in radicals. In Sections 4 and 5, formulas for roots of simple MVs are presented. Possible numerical methods of calculating the roots are discussed in Section 6, and finally, application of MV square root to solve Riccati equation in CA is demonstrated in Section 7.

Clifford algebra and notation
The Clifford algebra Cl p,q is an associative noncommutative algebra, where (p, q) indicates vector space metric. In 3D case, the MV consists of the following elements (basis blades) {1, e 1 , e 2 , e 3 , e 12 , e 13 , e 23 , e 123 ≡ I}, where e i are the orthogonal basis vectors, and e ij are the bivectors (oriented planes). The last term is called the pseudoscalar. The shorthand notation, for example, e ij means e ij = e i • e j , where the circle indicates the geometric or Clifford product. Usually, the multiplication symbol is omitted, and one writes e ij = e i e j . The number of subscripts indicates the grade of basis element, so that the scalar is a grade-0 element, the vectors are the grade-1 elements etc. In the orthonormalized basis, the products of vectors satisfies e i e j + e j e i = ±2δ ij . http://www.journals.vu.lt/nonlinear-analysis For Cl 3,0 and Cl 0,3 algebras, correspondingly, the squares of basis vectors are e 2 i = +1 and e 2 i = −1, where i = 1, 2, 3. For mixed signature algebras Cl 2,1 and Cl 1,2 , we have, respectively, e 2 1 = e 2 2 = 1, e 2 3 = −1 and e 2 1 = 1, e 2 2 = e 2 3 = −1. As a result, the sign of squares of blades depends on considered algebra. For example, in Cl 3,0 , we have e 2 12 = e 12 e 12 = −e 1 e 2 e 2 e 1 = −e 1 (+1)e 1 = −e 1 e 1 = −1. However, in Cl 1,2 , we have e 2 12 = −e 1 e 2 e 2 e 1 = −e 1 (−1)e 1 = e 1 e 1 = +1. The MV is a sum of different blades multiplied by real coefficients a A . In 3D algebras, the general MV expanded in coordinates (basis elements) reads A = a 0 + a 1 e 1 + a 2 e 2 + a 3 e 3 + a 12 e 12 + a 23 e 23 + a 31 e 31 + a 123 I where a = a 1 e 1 +a 2 e 2 +a 3 e 3 and B = a 12 e 12 +a 23 e 23 +a 31 e 31 represents, respectively, the vector and bivector parts of MV. The geometric product of two MVs A and B gives new multivector C = A • B, which has the same structure. In the following, the circle will be omitted, and AB will be interpreted as the geometric product of two MVs. Since we shall need the square of MV, below we present the full expression of A 2 in case of Cl 3,0 , where the scalar c 0 , vector c, bivector C and pseudoscalar c 123 I, c 123 ∈ R, expressed in components are ) − a 2 123 , c = 2(a 0 a 1 − a 23 a 123 )e 1 + 2(a 0 a 2 + a 13 a 123 )e 2 + 2(a 0 a 3 − a 13 a 123 )e 3 , C = 2(a 0 a 12 + a 3 a 123 )e 12 + 2(a 0 a 13 − a 2 a 123 )e 13 + 2(a 0 a 23 + a 1 a 123 )e 23 , c 123 = 2(a 0 a 123 + a 1 a 23 − a 2 a 13 + a 3 a 12 ). (3) Since we are considering real CAs, all coefficients a A , where A is the multiindex, are real numbers. For remaining algebras Cl 2,1 , Cl 1,2 and Cl 0,3 , the product formulas are similar to (3) except that signs before scalar coefficients may be different. Two main operations in CAs are the addition of MVs and their geometric product. Also, two supplementary operations -dot (or inner) and wedge (or outer) products -are frequently used in CA since they provide geometric interpretation to CA formulas and allow to introduce simpler notation. For example, in case of vectors a and b (italic notation, a and b, will be used too), the vector dot and wedge products can be expressed via geometric product respectively as a · b ≡ a · b = (ab + ba)/2 and a ∧ b ≡ a ∧ b = (ab − ba)/2. To select the scalar a 0 in MV (1), the grade selector A 0 = a 0 is introduced. In general, A k selects kth grade elements from the MV A, where k = 1, 2, 3 designates vector, bivector, or pseudoscalar. More about Clifford algebras and MV properties can be found, for example, in books [8,18]. If B and C are two MVs, due to noncommutativity of CA, we have that, in general, This property implies that the square root may have two signs. However, as we shall see in Section 4, this property is not general enough. In cases when there is an infinite number of roots, instead of ± √ B, we shall obtain more general (parameterized) expressions that also represent the square roots of B (see Sections 4 and 5).

Method of calculation
The method of calculation of MV square root is based on solution of equation A 2 = B, where A = a 0 + a 1 e 1 + · · · + a 12 e 12 + . . . and B = b 0 + b 1 e 1 + · · · + b 12 e 12 + · · · are the MVs in a fixed frame. Expanding A 2 (see Eqs. (2) and (3)) and equating scalar coefficients at the same grades of B, the system of real polynomial equations was constructed with respect to unknown coefficients {a 0 , a 1 , a 2 , . . . , a 12 , . . . , a 123 }. In case of 3D CAs, we have eight coupled real nonlinear equations. Real solutions (coefficients a A ) were found and analyzed with Mathematica package. The computer result consisted of a set of possible solutions (from 2 up to 28 depending on a number of nonzero coefficients in B). Since we consider real CAs only, the complex solutions (coefficients a A ) were rejected. The remaining real coefficients were grouped in plus/minus pairs ± √ B in case of isolated roots, or in alike pairs in case of infinite many roots. Finally, the results were checked both symbolically and numerically whether F 2 (B) = B is satisfied indeed. The numerical check allowed to eliminate spurious roots due to Mathematica specific manipulations with roots.

Example
As an illustration of the method, we will calculate the square root of MV in 1D, namely, in algebra Cl 0,1 , which is isomorphic to complex number algebra. The general MV in this algebra is A = a 0 + a 1 e 1 the square of which gives A 2 = a 2 0 − a 2 1 + 2a 0 a 1 e 1 . If B = b 0 +b 1 e 1 = A 2 is introduced, then MV equation A 2 = B yields two scalar equations: a 0 − a 1 = b 0 and 2a 0 a 1 = b 1 . Solution of this nonlinear system with respect to a 0 and a 1 gives four roots, where where B = b 0 +b 1 e 1 , and Û B = b 0 −b 1 e 1 is the grade inverse of B (an analogue of complex conjugate in the complex algebra). The grade inversion and inner product were used to represent the square root in the basis-free manner: 1 . If latter expressions are inserted back into (5), one finds the well-known formula for the square root of complex number. The role of imaginary unit in Cl 0,1 is played by basis vector e 1 since e 2 1 = −1. If b 0 = 0, then (5) reduces to becomes singular, and the system a 2 0 − a 2 1 = b 0 , 2a 0 a 1 = 0 is to be solved once more. Then the trivial root is This simple example shows that for those MVs that consist of individual grades (scalar, vector, etc) the roots must be calculated anew. As we shall see in Section 4.1.1, in higher dimensional noncommutative algebras, in such cases, there may appear free parameters that are connected with a continuum of roots.
The described method of finding square root was also applied to 2D Clifford algebras Cl 0,2 , Cl 1,1 and Cl 2,0 , which are isomorphic to quaternion, coquaternion and conecterine algebras [20]. We have found that in these algebras, there are square roots with additional free parameters that bring in a continuum of roots (unpublished results).

Square roots of general MV in 3D
The paper is devoted to investigation of square roots of blades and their simple combinations. Nonetheless, below we shall outline a constructive proof that in 3D algebras, the problem of square root of general multivector (MV) can be solved in radicals. Numerical solutions (see Section 6.1) show that at least four roots exist in case of general MV. In case of individual blades, as we shall see in Sections 4 and 5, infinitely many roots may appear, in a sharp contrast to general case where only isolated roots are allowed. The situation where there are infinitely many roots of quaternionic scalar has been noted earlier [15] and explained by existence of equivalence class in such roots. Thus, in CA, we have the situation where partial solutions do not necessarily follow from general solution, and, therefore, they must be considered to be as important as the general one. This is one of reasons why the present paper is confined to simple MVs, mainly to blade roots using computer-aided mathematics. Let us return back to roots of general MV.
Mathematica was unable to find the square root of the most general MV symbolically by the method described in Section 3.1. Below we shall show that, nonetheless, such a root does exist and even can be expressed in radicals. At first, we shall rewrite the initial MV A in terms of two scalars, s and S, and two vectors, v = v 1 e 1 + v 2 e 2 + v 3 e 3 and V = V 1 e 1 + V 2 e 2 + V 3 e 3 in the following form: The above system can be divided into subsystems (6) and (7)- (9). Subsystem (7)-(9) of six equations can be solved with respect to vector components It should be noted that the vectors v and V are functions of scalars s and S only. In this way obtained components of v and V are inserted back into scalar and pseudoscalar in Eqs. (6). This yields two coupled nonlinear algebraic equations for two unknowns s and S Since the highest degree of s and S in (13) is 5, the solution in radicals is not guaranteed to exist. However, the system can be easily solved numerically to give 4 isolated roots as demonstrated in Section 6.1. Fortunately, it appears that the change of variable s S = t and −s 2 + S 2 = T reduces system (13) to simpler one 2 , where the coordinate-free notation for the MV parts, has been introduced. The tilde is the index reversion operator, for example, e 12 = e 21 .
Since the substitution itself and the resulting system of equations with respect to t and T are both of degree 4, we conclude that the initial system (13) can be solved in radicals, i.e., an explicit formula of square root of MV when n 3 can be obtained. In particular, we can solve s and S from s S = t and −s 2 + S 2 = T , which yields two real solutions: In (15), the sign of both s i and S i should be the same. The other two solutions, which are obtained from (15) by the substitution (14) can be solved in radicals in a compact way as well. The two of solutions are real-valued due to the inequality D b S , where D = » b 2 S + b 2 I denotes the square root of determinant norm of MV B [2]. The other two solutions, which can be obtained from (16) by substitution D → −D, are complex valued.
To summarize, formulas (15), (16) and (6)-(12) yield four explicit real solutions and completely determine the square root A = s + v + (S + V)I = √ B in Cl 3,0 algebra in terms of radicals. Using the suggested transformations and notations, similar explicit formulas can be easily obtained for Cl 2,1 , Cl 1,2 and Cl 0,3 algebras too. More about square roots of a general MV can be found in [3].

Roots of simple MVs in Cl 3,0
Below the square roots of scalars, vectors, bivectors and pseudoscalars, and their combinations such as scalar-plus-vector and scalar-plus-pseudoscalar are calculated by the method described in Section 3.1 for algebras Cl 3,0 , Cl 2,1 , Cl 1,2 and Cl 0,3 .

Root of plus/minus scalar
Let A be Cl 3,0 algebra MV the square of which is either positive or negative scalar, A 2 = ±s and s ∈ R. Since A 2 is assumed to consist of all grades, from A 2 = ±s follows that the right-hand sides of resulting nonlinear system of equations are A 2 k = 0 if k = 1, 2, 3, and A 2 0 = ±s for scalar part. Such a system of eight equations is undetermined and allows free parameters in the solution. Using the method described in Section 3.1, below we will demonstrate that, apart from trivial roots, there may appear solutions that contain free parameters that represent infinitely many square roots. If solution method based on Gröbner basis (Section 3.1) is addressed, apart from these trivial expressions, one can find a more general formulas for square roots of plus/minus scalar that include up to four real free parameters √ s 1,2 = −αβδ ± wγ γ 2 + δ 2 e 1 + αβγ ± wδ γ 2 + δ 2 e 2 + αe 3 + βe 12 + γe 13 + δe 23 , The root √ s 1 or √ s 2 , respectively, corresponds to upper and lower signs. The formula is valid when s 1,2 > 0 and s 1,2 < 0. The Greek letters α, β, γ, δ ∈ R are optional real parameters, which, in fact, represent MV coefficients that appeared redundant in solving the system of polynomial equations: a 3 → α, a 12 → β, a 13 → γ, a 23 → δ.
It can be checked that after squaring of the right side of (17), one gets ( √ s 1 ) 2 = √ s 1 × √ s 1 = s and ( √ s 2 ) 2 = √ s 2 √ s 2 = s, i.e., after squaring, all free parameters will cancel out automatically. Thus, the appearance of free parameters allows Eq. (17) to have an infinite number of square roots. Also, one should note that the structure of the roots in (17) is different from simple expression ± √ B when parameters are absent. If all four parameters are equated to zero in an appropriate order to avoid the infinity, one gets simple expression, either + √ s e 1 and + √ s e 2 , or − √ s e 1 and − √ s e 2 , the squares of which give the initial scalar s. More interesting cases are those where either three, two or single of parameters are equated to zero.
Single real parameter Here the roots lie on hyperbolas in parameter space (± s + γ 2 , γ) or (± √ s + δ 2 , δ) with branches pointing to left and right.
Three real parameters Finally, we shall note that in case of quaternionic root of a scalar, there appear two free parameters [unpublished results], and the structure of root formula is very similar to Cl 3,0 , namely, √ s = ± s − α 2 − β 2 e 1 + αe 2 + βe 12 . If basis vectors of Cl 0,2 algebra are replaced by quaternionic imaginary units (e 1 → k, e 2 → j, e 12 → i) and free parameters are replaced by angles β = √ s sin θ cos ϕ and α = √ s sin θ sin ϕ, the formula for quaternionic root of scalar can be reduced to √ s(± cos θ k + sin θ sin ϕ j + sin θ cos ϕ i), which represents the equation of sphere. Thus, in the quaternionic case, we have a continuum of roots on sphere of radius √ s [15].

Square root of plus/minus scalar in Cl 0,3
In contrast to Euclidean Cl 3,0 algebra, in case of anti-Euclidean Cl 0,3 algebra, the square root of scalar was found to have only trivial solutions. No free parameters have been detected. When scalar s > 0, the roots are √ s = ± √ s, √ s = ± √ s I, s > 0.
The second root comes from pseudoscalar property I 2 = 1 in Cl 0,3 . More interesting is that in Cl 0,3 , the square root of negative scalar, i.e., ± √ −s when s > 0, does not exist at all, i.e., no MV solutions with real coefficients have been found by algorithm described in Section 3.1. Table 1 summarizes all possible square roots of vector a = a 1 e 1 + a 2 e 2 + a 3 e 3 in 3D Clifford algebras. I is the pseudoscalar of a respective algebra. The value of a 2 depends on algebra: a 2 = ±(a 2 1 + a 2 2 + a 2 3 ) in Cl 3,0 (upper sign) and Cl 0,3 (lower sign); a 2 = a 2 1 − a 2 2 − a 2 3 in Cl 1,2 and a 2 = a 2 1 + a 2 2 − a 2 3 in Cl 2,1 . One can see that, in general, there are more then two square roots of vector; the largest number (up to 8) is in Cl 1,2 . The free parameters do not appear. Table 1. Square roots of vector a = a 1 e 1 + a 2 e 2 + a 3 e 3 , a i ∈ R, in 3D Clifford algebras. The numerical value and sign of a 2 depends on algebra and vector coefficients a i . The root √ a belongs to real Clifford algebra if expression under the square root in formulas is positive. The square root of e 1 (a 2 > 0) does not exist. Since in this algebra I 2 = 1 > 0, the square root of pseudoscalar I (a 2 > 0) does not exist too. In this algebra, the square roots of general vector a = e 1 + e 2 /2 + √ 3 e 3 (a 2 = −7/4 < 0) are √ a = ±( √ 7 + 2(e 1 + e 2 /2 + √ 3 e 3 ))/(2 · 7 1/4 ) and √ a = ±( √ 7 + 2(e 1 + e 2 /2 + √ 3 e 3 ))I/(2 · 7 1/4 ). Table 2 shows that in 3D algebras, the bivector can have two (Cl 3,0 ), four (Cl 0,3 and in Cl 2,1 . The free parameters do not appear. Table 2. Square roots of bivector B = a 12 e 12 + a 13 e 13 + a 23 e 23 , a ij ∈ R in 3D Clifford algebras.

Roots of bivector
Algebra Square root of bivector B Constraint

Roots of pseudoscalar
Only square roots in Euclidean algebra Cl 3,0 is considered here. It should be remembered that in MV equation A 2 = B, the MV A takes into account all gradings, A = a 0 + a 1 e 1 + a 2 e 2 + a 3 e 3 + a 12 e 12 + a 13 e 13 + a 23 e 23 + a 123 I, whereas the root of B consists of pseudoscalar Mathematica command Solve[..] finds 28 possible candidates for roots of A 2 = B, the most of which have complex coefficients and, thus, must be rejected. Only four of roots have real coefficients. Similarly as in the scalar case, those coefficients of A that survive in the final result will be treated as free parameters. All in all, we have found that five of coefficients in A may be considered as free parameters: ε = a 2 , α = a 3 , β = a 12 , β = a 12 , γ = a 13 , and δ = a 23 . The root has no free parameters. The second root has two free parameters β and ε, The third root has three free parameters α, β, γ, Finally, the fourth root has four free parameters α, β, δ, γ, where Formulas (18)- (21) are valid when p > 0 and p < 0, i.e., for roots of positive and negative pseudoscalar. Depending on sign of p, it can be checked that the squares of right-hand sides of (18)-(21) simplify to either +pI or −pI.

Square roots of paravector
The electromagnetism theory is inherently relativistic and requires 4D algebra Cl 1,3 for its full description [8]. Nonetheless, the covariant formulation of electromagnetism can be introduced if paravectors of smaller Cl 3,0 algebra are addressed [4]. The paravector is a sum of scalar and vector. The geometric product of paravector, p = a 0 + a, and its grade conjugate, Û p = a 0 − a, gives scalar pÛ p = a 2 0 − a 2 , which is just the Minkowski space-time metric if a 0 is interpreted as time component.
We have found that in Cl 3,0 , the paravector has 12 square roots as summarized in Table 3. The roots assume two different forms. In all formulas in Table 3, the expressions under square roots must be positive for MV coefficients to be real. Table 3. Square roots of paravector p = s + a, where s is the scalar and a is the vector.
Algebra Square root of paravector p = s + a Constraint

Square root of scalar + pseudoscalar
In Cl 3,0 , the square roots of a = s + pI, where s, p ∈ R, in a simple (nonparameterized) form are √ a = s + pI where s = a 0 and p = −aI 0 . In the parameterized form, the expressions for square roots may have two, three or four parameters that represent the coefficients of A, namely, µ = a 2 , α = a 3 , β = a 12 , γ = a 13 , δ = a 23 .
The square root with two free parameters β and µ is The square root with three free parameters α, β and γ is The square root with four free parameters α, β, δ and γ is The four-parameter formula depicts a hypersurface in five dimensional parameter space, where infinite number of roots exists. After squaring of the right-hand side of (25), all parameters in the formula cancel out to yield a = s + pI. If α = β = γ = 0, formula (25) reduces to √ a = pe 1 ± −p 2 + 4δ 2 (s + δ 2 ) e 2 2δ + δe 23 , where, as always, the expression under the square root must be positive. The formula has singularity at δ = 0. After squaring, the singularity disappears. The presence of vector e 1 and bivector e 23 ensures that after squaring of the right-hand side the elementary pseudoscalar I = e 123 will appear in a = s + pI.

Square root by series expansion
The standard series formula of square root is or, in short, where k!! = k(k−2)(k−4) · · · . We shall generalize (26) and (27) to MV case. Now, 1 and x are replaced by, respectively, scalar and remaining part of MV. If A is the multivector, then in the Clifford algebra, formula (26) becomes where A is the scalar part, B = A/ A − 1 and √ 1 + B is MV series similar to (26), The condition x 2 1 now must be replaced by condition for norm B 1. For example, let us assume that A = 1 + e 1 /2 + e 12 /2 + e 123 /3. If series (28) is summed up to B 10 , then in Cl 3,0 , we find that After squaring the right-hand side of (29), we have A ≈ 1. + 0.500001e 1 + 0.500001e 12 + 1.95 · 10 −7 e 3 − 1.95 · 10 −7 e 23 + 0.333333e 123 , which shows that the error is in the seventh digit, and therefore, the terms with e 3 and e 23 must be rejected. The norm of MV under square root is 20/18 ≈ 1.27, which is somewhat larger than unity. To reduce the total number of MV multiplications, it may be useful to rewrite the MV series in the Horner form: ããã .

Newton's iteration method
As known, the square root of a real number a can be effectively calculated by Newton's iteration formula, √ a → x k+1 = (x k + a/x k )/2. The iteration frequently starts with x 0 = a. This formula works in matrix form as well, albeit with some modifications to improve the convergence [11]. Here we present the first attempt to see whether, in principle, the Newton's method is applicable to CA. If multiplication is replaced by MV multiplication, the Newton's formula can be written in CA in the following form: X 0 is an initial MV, k = 0, in the iteration. Also, the term X −1 k A may be replaced by AX −1 k . Numerical experiments show that Newton-Clifford formula (30) works as well. To avoid divergence during iteration, it is recommended the scaling, which consists of division of all MV coefficients by the largest coefficient c and multiplication of the final result by √ c. As an example, let us find the square root of A = e 1 + 2e 3 + 2e 12 + e 23 , A ∈ Cl 3,0 .
After 12 iterations, we obtain √ A ≈ 1.16172+0.215199e 1 +0.430397e 3 +1.03907e 12 + 0.519536e 23 − 0.481199e 123 , the squaring of which returns back the MV A with a high precision. However, the Newton's algorithm comes to a vicious cycle if one tries to find the square root of a pure vector. To break the circle, we have added a small seed bivector to initial MV X 0 . For example, we have assumed that X 0 = A + e 12 /5, where now A is the vector A = e 1 + 3e 2 + 2e 3 . Then, the Newton's algorithm after 12 iterations and with a scaling factor 4 (division of A by 4 and multiplication of Thus, after 12 Newton's iterations, the error spreads out from the third significant digit. Further investigations are needed to understand the stability and convergence properties of the Newton-Clifford iterator. Also, one should remember that the MV may have multiple roots. The investigations of convergence of roots in matrix theory [11] are mainly concerned with matrices having special spectrum, which must lie in the real half of a complex plane of eigenvalues to guarantee the fast convergence [11]. On the other hand, the matrices that represent Clifford algebra are totally different. Their properties are described by 8-periodicity table [18] rather than matrix spectrum and, as a consequence, the totally different methods that take into account the periodicity are required to investigate convergence properties of Newton-Clifford algorithm.

Application: Algebraic Riccati equation
The matrix Riccati equation is frequently addressed in control and systems theory [1,16,17]. Solution of quaternionic (Cl 0,2 ) Riccati equation has been presented in [16]. Here we shall consider the Riccati-Clifford equation where matrices are replaced by MVs and matrix product by geometric (Clifford) product. The simplest quadratic CA equations are where A and B are known and X is unknown MV. The solutions of (31) can be expressed through the square root and inversion of MV, The validity of the second solution can be easily verified in the following way: A simple Riccati-Clifford equation with a linear term added, can be solved in a similar manner. After multiplication by A from right and addition of (p/2) 2 to both sides, Eq. (32) reduces to the solution of which is A slightly general Riccati-Clifford equation can be solved too if MV C belongs to the center of geometric algebra, i.e., C commutes with X. For example, in Cl 3,0 algebra, the center is a sum of scalar and pseudoscalar, C = α + βI, α, β ∈ R. Then, one can check that the solution of Eq. (33) is If MVs are replaced by scalars (A → a, B → c and C → b/2) and geometric product by standard product, then (34) reduces to the well-known scalar quadratic equation for roots, x 1,2 = (−b ± √ b 2 + 4ac)/2a. Thus, to solve the Clifford-Riccati equation one must know MV square root. How to calculate the inverse MV A −1 in (34), recently it was described in [2]. It should be noted that the solution (34) remains valid for an arbitrary CA if C is the center of algebra.
Insertion of X back into (33) shows that the Riccati-Clifford equation is satisfied indeed. A more interesting situation arises when free parameters are allowed in the square root. For example, if we assume that µ = 0 in (23), then we have the following square root with a single parameter β: √ s + pI = + s − (p/2β) 2 + β 2 e 1 + (p/2β)e 3 + βe 12 . Then, the solution of Riccati-Clifford equation (34)  This one-parameter solution also satisfies the Riccati-Clifford equation (33). For a solution to exist, one must ensure that the MV coefficients are real, i.e., the free parameter must satisfy −44 + 4β 2 − 961/β 2 0 or β (11/2 + 541/2 ) 1/2 . Similarly, using Eqs. (22)-(25), one can find the solutions of algebraic Riccati-Clifford equation with two, three, or even with four free parameters.

Conclusions
In the present article, the Gröbner-basis algorithm was applied to extract the square root from MV in real algebras (CA) Cl 3,0 , Cl 2,1 , Cl 1,2 and Cl 0,3 . It is shown that the square root of general MV can be expressed in radicals, and there are four isolated roots in mentioned algebras. In this article, the main attention, however, was concentrated on square roots of individual grades of MV or their combinations. A number of concrete square root formulas for scalars, vectors, bivectors and pseudoscalars as well as scalarvector and scalar-pseudoscalar combinations are presented in a basis-free form. It is shown that roots of graded MVs may accommodate a number free parameters that bring a continuum of square roots on respective parameter hypersurfaces and therefore become an important ingredient in the nonlinear problems. Concrete expressions that contain up to four free parameters are presented. The free parameters appear in those MVs, which belong to the center of algebra, and therefore, the MVs make up an equivalence class in the considered algebra. Also, a number of numerical methods to calculate square roots in CAs are demonstrated. However, more investigations are required to optimize and adapt them in calculations of multiple roots. Conversely to matrix square roots [11], where the spectrum of matrix must be positive for a square root to exist, in CA where the MV can be represented by matrices too (although of special form as follows from the 8-periodicity table [18]), no such requirement is needed. The Riccati-Clifford equation was solved in CA terms, which is expected to pave the way to control theory based on more-and-more popular CA.