A Solution Approach for Lower Hessenberg Linear Systems

Abstract

An approach is developed to obtain solutions to lower Hessenberg linear systems with general entries. The approach involves developing solution vectors for an extended lower Hessenberg linear system (having an extra column and an extra introduced unknown) for each nonzero term on the right hand side. The overall solution is then found through superposition and determination of the extra introduced unknown. The approach supports parallel solution algorithms without communication between processors, since each solution vector is computed independently of the others. The number of parallel processors needed will be equal to the number of nonzero right hand side terms.

1. Introduction

A number of researchers have studied the inverses of lower Hessenberg matrices [1–4], that is, inverses of square matrices of the form 𝐴=(𝑎_𝑖,𝑗), 𝑖,𝑗=1,…,𝑛 such that 𝑎_𝑖,𝑗=0 when 𝑗>𝑖+1. Most recently, a recursive algorithm has been developed for inverting Hessenberg matrices [1]. This paper proposes an alternate solution approach. It is shown here that lower Hessenberg linear systems in particular lend themselves to a solution via an extended system (that adds a column to 𝐴, as well as an additional unknown). The basic strategy involves generating solution vectors for 𝑚 such extended systems (where 𝑚 is the number of nonzero right hand side terms). In each such extended system, all of the right hand side terms are set to zero except for one entry. The first term of each solution vector is chosen arbitrarily, and then each subsequent term is found directly through a forward-substitution-like process similar to that used in LU decomposition, with a number of operations on the order of that required for forward substitution, that is, an order of magnitude smaller than that required for performing the LU decomposition itself. The overall solution is found through superposition and solution of the extra introduced variable (which is common to all of the 𝑚 extended systems). The process is highly parallelizable, since each solution vector can be computed independently of the others.

This approach will first be illustrated for a three-dimensional system (i.e., 𝑛=3) with general entries in Section 2. Section 3 provides a proof of the validity of the approach for systems having an arbitrary dimension 𝑛.

2. Three-Dimensional Systems

This section details the approach to solve a lower Hessenberg linear system with 𝑛=3 and general entries. Consider the following system: ⎡⎢⎢⎢⎢⎣𝑎𝑎_1,12,1𝑎_1,20𝑎_2,2𝑎_2,3𝑎_3,1𝑎_3,2𝑎_3,3⎤⎥⎥⎥⎥⎦⎧⎪⎪⎨⎪⎪⎩𝑥₁𝑥₂𝑥₃⎫⎪⎪⎬⎪⎪⎭=⎧⎪⎪⎨⎪⎪⎩𝑏₁𝑏₂𝑏₃⎫⎪⎪⎬⎪⎪⎭.(2.1)

This can be solved via an extended system developed by adding an additional column to the coefficient matrix, and an additional unknown 𝑥₄:⎡⎢⎢⎢⎢⎣𝑎𝑎_1,1𝑎_1,200_2,1𝑎_3,1𝑎_2,2𝑎_2,30𝑎_3,2𝑎_3,31⎤⎥⎥⎥⎥⎦⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑥¹𝑥₂𝑥₃𝑥₄⎫⎪⎪⎪⎬⎪⎪⎪⎭=⎧⎪⎪⎨⎪⎪⎩𝑏₁𝑏₂𝑏₃+𝑥₄⎫⎪⎪⎬⎪⎪⎭,(2.2) where 𝑎_𝑖,𝑖+1≠0 for all 𝑖. The coefficient 𝑎_3,4 is set to 1 to simplify the solution process. Note that the third equation in (2.2) is 𝑎_3,1𝑥₁+𝑎_3,2𝑥₂+𝑎_3,3𝑥₃+𝑥₄=𝑏₃+𝑥₄,(2.3) or 𝑎_3,1𝑥₁+𝑎_3,2𝑥₂+𝑎_3,3𝑥₃=𝑏₃.(2.4)

The strategy for solving (2.2) involves first solving three such systems, each with a different nonzero right hand side term to obtain the solution vectors 𝑣⁽¹⁾, 𝑣⁽²⁾, and 𝑣⁽³⁾, and then using superposition. The first such system, ⎡⎢⎢⎢⎢⎣𝑎𝑎_1,1𝑎_1,200_2,1𝑎_3,1𝑎_2,2𝑎_2,30𝑎_3,2𝑎_3,31⎤⎥⎥⎥⎥⎦⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑣¹⁽¹⁾𝑣²⁽¹⁾𝑣³⁽¹⁾𝑣⁴⁽¹⁾⎫⎪⎪⎪⎬⎪⎪⎪⎭=⎧⎪⎪⎨⎪⎪⎩100⎫⎪⎪⎬⎪⎪⎭,(2.5) is solved by choosing 𝑣¹⁽¹⁾ arbitrarily and then determining the remaining terms using the subsequent equations in (2.5). Setting 𝑣¹⁽¹⁾=𝑎_1,2𝑎_2,3 leads to 𝑣²⁽¹⁾=1𝑎_1,2𝑣₃₍₁₎−𝑎_1,1𝑎_2,3,=−𝑎^2,2𝑎_1,2𝑎_2,3𝑣₄₍₁₎+𝑎_1,1𝑎_2,2−𝑎_1,2𝑎_2,1,=𝑎^3,3𝑎_2,2𝑎_1,2𝑎_2,3−𝑎^3,2𝑎_1,2+𝑎_1,1𝑎_2,3𝑎_3,2−𝑎_2,2𝑎_3,3+𝑎_1,2𝑎_2,1𝑎_3,3−𝑎_2,3𝑎_3,1.(2.6)

This process can be continued to compute all of the 𝑣^𝑗(𝑘) terms. The next system to be considered is: ⎡⎢⎢⎢⎢⎣𝑎𝑎_1,1𝑎_1,200_2,1𝑎_3,1𝑎_2,2𝑎_2,30𝑎_3,2𝑎_3,31⎤⎥⎥⎥⎥⎦⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑣¹⁽²⁾𝑣²⁽²⁾𝑣³⁽²⁾𝑣⁴⁽²⁾⎫⎪⎪⎪⎬⎪⎪⎪⎭=⎧⎪⎪⎨⎪⎪⎩010⎫⎪⎪⎬⎪⎪⎭.(2.7)

Setting 𝑣¹⁽²⁾=𝑎_1,2𝑎_2,3 leads to 𝑣𝑣²⁽²⁾₃₍₂₎=−𝑎_1,1𝑎_2,3,=1𝑎_2,3𝑣₄₍₂₎+𝑎_1,1𝑎_2,2−𝑎_1,2𝑎_2,1,=−𝑎^3,3𝑎_2,3+𝑎_1,1𝑎_2,3𝑎_3,2−𝑎_2,2𝑎_3,3+𝑎_1,2𝑎_2,1𝑎_3,3−𝑎_2,3𝑎_3,1.(2.8)

The third system is: ⎡⎢⎢⎢⎢⎣𝑎𝑎_1,1𝑎_1,200_2,1𝑎_3,1𝑎_2,2𝑎_2,30𝑎_3,2𝑎_3,31⎤⎥⎥⎥⎥⎦⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑣¹⁽³⁾𝑣²⁽³⁾𝑣³⁽³⁾𝑣⁴⁽³⁾⎫⎪⎪⎪⎬⎪⎪⎪⎭=⎧⎪⎪⎨⎪⎪⎩001⎫⎪⎪⎬⎪⎪⎭.(2.9)

Setting 𝑣¹⁽³⁾=𝑎_1,2𝑎_2,3 leads to 𝑣𝑣²⁽³⁾₃₍₃₎=−𝑎_1,1𝑎_2,3,𝑣₄₍₃₎=𝑎_1,1𝑎_2,2−𝑎_1,2𝑎_2,1,=1+𝑎_1,1𝑎_2,3𝑎_3,2−𝑎_2,2𝑎_3,3+𝑎_1,2𝑎_2,1𝑎_3,3−𝑎_2,3𝑎_3,1.(2.10)

Superposition of the vectors 𝑣⁽¹⁾, 𝑣⁽²⁾, and 𝑣⁽³⁾ leads to a solution for (2.2), that is, ⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑥₁𝑥₂𝑥₃𝑥₄⎫⎪⎪⎪⎬⎪⎪⎪⎭=𝑏₁⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑣¹⁽¹⁾𝑣²⁽¹⁾𝑣³⁽¹⁾𝑣⁴⁽¹⁾⎫⎪⎪⎪⎬⎪⎪⎪⎭+𝑏₂⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑣¹⁽²⁾𝑣²⁽²⁾𝑣³⁽²⁾𝑣⁴⁽²⁾⎫⎪⎪⎪⎬⎪⎪⎪⎭+𝑏₃+𝑥₄⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑣¹⁽³⁾𝑣²⁽³⁾𝑣³⁽³⁾𝑣⁴⁽³⁾⎫⎪⎪⎪⎬⎪⎪⎪⎭.(2.11)

From (2.11), note that 𝑥₄=𝑏₁𝑣⁴⁽¹⁾+𝑏₂𝑣⁴⁽²⁾+𝑏₃+𝑥₄𝑣⁴⁽³⁾.(2.12)

Substituting into (2.12) for 𝑣⁴⁽¹⁾, 𝑣⁴⁽²⁾, and 𝑣⁴⁽³⁾ (and performing some algebra) yields 𝑥₄=−𝑏₁−𝑏₂−𝑏₃+𝑏¹det(𝐴)𝑎^3,3𝑎_2,2𝑎_1,2𝑎_2,3−𝑎^3,2𝑎_1,2−det𝑏²(𝐴)𝑎𝑎^3,3_2,3+𝑏³det(𝐴),(2.13) where det(𝐴)=𝑎_1,1𝑎_2,2𝑎_3,3−𝑎_2,3𝑎_3,2+𝑎_1,2𝑎_2,3𝑎_3,1−𝑎_2,1𝑎_3,3.(2.14)

Finally, substituting 𝑥₄ and 𝑣⁽¹⁾, 𝑣⁽²⁾, and 𝑣⁽³⁾ into (2.11) leads to the solution to (2.1), after some algebra: 𝑥₁=𝑏¹𝑎_2,2𝑎_3,3−𝑎_2,3𝑎_3,2−𝑏₂𝑎_1,2𝑎_3,3+𝑏₃𝑎_1,2𝑎_2,3𝑥det(𝐴)₂,=𝑏¹𝑎_2,3𝑎_3,1−𝑎_2,1𝑎_3,3+𝑏₂𝑎_1,1𝑎_3,3−𝑏₃𝑎_1,1𝑎_2,3𝑥det(𝐴)₃,=𝑏¹𝑎_2,1𝑎_3,2−𝑎_2,2𝑎_3,1+𝑏₂𝑎_1,2𝑎_3,1−𝑎_1,1𝑎_3,2+𝑏₃𝑎_1,1𝑎_2,2−𝑎_2,1𝑎_1,2det(𝐴).(2.15)

The next section extends the process for a system having an arbitrary dimension 𝑛. However, few comments are first warranted. It can be inferred by inspection of (2.11) that computations are minimized when 𝑚, the number of nonzero right hand side terms, is small. For example, when 𝑚=1, only two of the 𝑣^(𝑘) vectors need to be computed, regardless of 𝑛. (Only one vector needs to be computed if 𝑏_𝑛 is the only nonzero right hand side term.) The number of operations to compute a single vector 𝑣^(𝑘) is on the order of that for the forward substitution process used in LU decomposition, or an order of magnitude smaller than performing the LU decomposition process itself. Thus, the approach can be useful when 𝑚≪𝑛. However, even when 𝑚≈𝑛, the solution for the 𝑣^(𝑘) vectors can be performed on 𝑚 parallel processors, since each 𝑣^(𝑘) can be determined simultaneously on a single processor without any communication needed with any of the other processors. Thus, the run time in such a parallel algorithm should approach that for computing just one of the 𝑣^(𝑘) vectors.

A considerable amount of recent work has involved attempts to parallelize the solutions of linear systems [5–7] (e.g., involving LU or QR decompositions). Typically, the degree of parallelism achieved is only partial and is highly dependent on the structure and sparseness of 𝐴 and requires varying degrees of communication between processors. In contrast, this approach provides a framework for full parallelization with 𝑚 processors without any communication required between processors.

3. Proof of the General Case

This section provides a proof that the algorithm from the previous section is valid for the general case, that is, a lower Hessenberg linear system having arbitrary dimension 𝑛. Consider the system of equations ^𝑛_𝑗=1𝑎_𝑖,𝑗𝑥_𝑗=𝑏_𝑖,(3.1) for each value of 𝑖 where 1≤𝑖≤𝑛, and 𝑎_𝑖,𝑗=0 when 𝑗>𝑖+1. Following the procedure in the last section, consider the extended system ^𝑖+1_𝑗=1𝑎_𝑖,𝑗𝑥_𝑗=𝑏_𝑖+𝛿_𝑖,𝑛𝑥_𝑛+1,(3.2) for each value of 𝑖, where 1≤𝑖≤𝑛. Here 𝑎_𝑖,𝑗=0 when 𝑗>𝑖+1, 𝛿_𝑖,𝑛 is the Kronecker delta, 𝑎_𝑛,𝑛+1=1, and 𝑎_𝑖,𝑖+1≠0 for all 𝑖. Consider further the 𝑛 related systems of the form ^𝑖+1_𝑗=1𝑎_𝑖,𝑗𝑣^𝑗(𝑘)=𝛿_𝑘,𝑖,(3.3) for each 𝑘, where 1≤𝑘≤𝑛, and for each 𝑖, where 1≤𝑖≤𝑛.

For a given 𝑘, each successive term of 𝑣^(𝑘) can thus be obtained by solving a single equation of the system (3.3) for one unknown (similar to the forward substitution process used in LU decomposition).

The 𝑛+1 solutions 𝑣^(𝑘) for 𝑘=1,…,𝑛, once obtained, can then be superimposed as follows for 𝑖≤𝑛+1: 𝑥_𝑖=^𝑛_𝑘=1𝑏_𝑘𝑣^𝑖(𝑘)+𝑥_𝑛+1𝑣^𝑖(𝑛).(3.4)

Equation (3.4) satisfies (3.2) and provides an equation to determine 𝑥_𝑛+1, that is, 𝑥_𝑛+1=^𝑛_𝑘=1𝑏_𝑘𝑣^(𝑘)𝑛+1+𝑥_𝑛+1𝑣^(𝑛)𝑛+1.(3.5)

Since (3.4) satisfies (3.2) and 𝑏₁,𝑏₂,…,𝑏_𝑛 are arbitrary, it follows that (3.5) leads to a solution for 𝑥_𝑛+1. Substituting 𝑥_𝑛+1 obtained from (3.5) into (3.4) then determines all of the 𝑥_𝑖 terms. Finally, the 𝑛th equation of (3.2) is ^𝑛_𝑗=1𝑎_𝑛,𝑗𝑥_𝑗+𝑥_𝑛+1=𝑏_𝑛+𝑥_𝑛+1,(3.6) or ^𝑛_𝑗=1𝑎_𝑛,𝑗𝑥_𝑗=𝑏_𝑛,(3.7) which is identical to the 𝑛th equation in (3.1). Thus, this approach solves (3.2) and also solves (3.1), since (3.1) is a subset of (3.2) and is further isolated from the additional variable 𝑥_𝑛+1.

4. Concluding Remarks

It has been shown that lower Hessenberg linear systems can be solved by considering related extended systems, first by directly solving a system with 𝑛=3, and then extending the procedure to arbitrary 𝑛. The approach involving the development of 𝑚 solution vectors from an extended system (having an additional column in the coefficient matrix and an additional introduced unknown) is highly parallelizable on 𝑚 processors.

It should be noted that this approach also lends itself to lower 𝑞-Hessenberg linear systems, that is, involving square matrices 𝐴=(𝑎_𝑖,𝑗), 𝑖,𝑗=1,…,𝑛 such that 𝑎_𝑖𝑗=0 when 𝑗>𝑖+𝑞. For example, 𝑞=2 leads to two additional columns and two additional introduced variables 𝑥_𝑛+1 and 𝑥_𝑛+2, that are solved with two additional equations. The proof is similar.

There is no limit in principle to the number of diagonals that can be added. Each will lead to an additional introduced unknown and an additional column to 𝐴. The process can thus be extended to many systems of practical interest, providing a framework for parallelization of such systems.