This tutorial discusses Householder reduction of n linear equations to a triangular form which can be solved by back substitution. The main strengths of the method are its numerical stability and suitability for parallel computing. We explain how Householder reduction can be derived from elementary matrix algebra. The method is illustrated by a numerical example and a Pascal algorithm. We assume that the reader has a general knowledge of vector and matrix algebra but is less familiar with linear transformation of a vector space

Hansen, Per Brinch

English

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Electrical Engineering and Computer Science - Technical Reports College of Engineering and Computer Science 12-1990 HOUSEHOLDER REDUCTION Per Brinch Hansen Syracuse University, School of Computer and Information Science, pbh@top.cis.syr.edu Follow this and additional works at: https://surface.syr.edu/eecs_techreports  Part of the Computer Sciences Commons Recommended Citation Hansen, Per Brinch, "HOUSEHOLDER REDUCTION" (1990). Electrical Engineering and Computer Science - Technical Reports. 78. https://surface.syr.edu/eecs_techreports/78 This Report is brought to you for free and open access by the College of Engineering and Computer Science at SURFACE. It has been accepted for inclusion in Electrical Engineering and Computer Science - Technical Reports by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. HOUSEHOLDER REDUCTION PER BRINCH HANSEN December 1990 School of Computer and Information Science Suite 4-116 Center for Science and Technology Syracuse, New York 13244-4100 (315) 443-2368 SU-CIS-90-39 HOUSEHOLDER REDUCTION1 PER BRINCH HANSEN School of Computer and Information Science Syracuse University Syracuse, New York 13244 December 1990 Abstract. This tutorial discusses Householder reduction of n linear equations to a triangular form which can be solved by back substitution. The main strengths of the method are its numerical stability and suitability for parallel computing. \Ve explain how Householder reduction can be derived from elementary matrix algebra. The method is illustrated by a numerical example and a Pascal algorithm. vVe assume that the reader has a general knowledge of vector and matrix algebra but is less familiar with linear transformation of a vector space. Key words. Linear systems, Householder reduction. AMS (MOS) subject classifications. 65F05 (direct methods for linear sys-tems, Householder reduction). Introduction. The solution of linear equations is important in many areas of sci-ence and engineering (Kreyszig [1988]). This tutorial discusses Householder reduction of n linear equations to a triangular form which can be solved by back substitution (Householder [1958], Press (1989]). The main strengths of the method are its nu-merical stability and suitability for parallel computing (Ortega [1988], Brinch Hansen [1990]). Text books on numerical analysis often produce Householder reduction like a rabbit from a magician's top hat. vVe will explain how the method can be derived from elementary matrix algebra. The method is illustrated by a numerical example and a Pascal algorithm. We assume that the reader has a general knowledge of vector and matrix algebra but is less familiar with linear transformation of a vector space. We begin by looking at the problems of Gaussian elimination. 1Copyright@l990 Per Brinch Hansen HOUSEHOLDER REDUCTION 2 1. Gaussian elimination. The classical method for solving a system of linear equations is Gaussian elimination. Suppose we have three linear equations with three unknowns xll Xz, x3: 4x3 - 18 2x3 - 1 + 3x3 - 14 First we eliminate x1 from the second equation by subtracting 1/2 of the first equation from the second one. Then we eliminate Xt from the third equation by subtracting 3/2 of the first equation from the third one. We now have three equations in which x1 occurs in the first equation only 2xt + 2xz + 4x3 - 18 2xz 4x3 - -8 2xz 3x3 - -13 Finally we eliminate x 2 from the third equation by adding the second equation to the third one. The equations have now been reduced to a triangular form which has the same solution as the original equations but is easier to solve 2x1 + 2xz + 4x3 - 18 2xz 4x3 - -8 7x3 - -21 The triangular equations are solved by back substitution. From the third equation we immediately have x3 = 3. By substituting this value in the second equation, we find x 2 = 2. Substituting these two values in the first equation we obtain x1 = 1. In general we haven linear equations with n unknowns (1.1) auXt + a12xz + az1X1 + azzXz + + a1nXn = b1 + aznXn = b2 The a's and b's are known real numbers. The x's are the unknowns we must find. The equation system (1.1) can be expressed as a vector equation (1.2) Ax = b where A is an n X n matrix, while x and bare n-dimensional column vectors au a12 a1n Xt bl a21 a22 a2n X2 b2 A= x= b= Xn bn HOUSEHOLDER REDUCTION 3 The equation system has a unique solution only if the matrix A is non-singular as defined in the appendix. Gaussian elimination reduces Eq. (1.2) to an equivalent form Ux = c where U is an n x n upper triangular matrix Un U12 Utn 0 U22 U2n U= 0 0 with all zeros below the main diagonal, while cis an n-dimensional column vector. Gaussian elimination requires O(n3 ) operations. The scaling of equations is a source of numerical errors in Gaussian elimination. To eliminate the ith unknown from the Ph equation we subtract the ith equation multiplied by aid aii from the Ph equation. If the pivot element aii is very small, the scaling factor becomes very large and we may end up subtracting very large reals from very small ones. This makes the results highly inaccurate. The numerical instability of Gaussian elimination can be reduced by pivoting, a rearrangement of the rows and columns which makes the pivot element as large as possible. On a parallel computer pivoting complicates the elimination algorithm (Fox [1988]). In the following we describe an alternative method which is numerically stable and does not require pivoting. 2. Scalar products. Householder reduction of an n x n real matrix has a simple geometric interpretation: The matrix columns are regarded as vectors in an n-dimensional space. Each vector is replaced by its mirror image on the other side of a particular plane. This plane reflects the first column onto the first axis of the coordinate system to produce a new column with all zeros after the first element. Householder's method requires the computation of scalar products and vector reflections. The following is a brief explanation of these basic operations. The ap-pendix defines the elementary laws of vector and matrix algebra, which we will take for granted. Let a and b be two n-dimensional column vectors al bl a2 b2 a= b= an bn The transpose of a and b are the row vectors aT = [at a2 ... an] bT = [bt b2 ... bn] HOUSEHOLDER REDUCTION 4 The scalar product of a and b is (2.1) A scalar product is obviously symmetric (2.2) The Euclidean norm (2.3) is the length of an n-dimensional vector a. From Eqs. (2.1) and (2.3) we obtain an equivalent definition of the norm (2.4) 3. Reflection. Figure 1 shows a unit vector v in 3-dimensional space. The dotted line represents a plane P which is perpendicular to v through the origin 0. For an arbitrary vector a we wish to find another vector b, which is the reflection of a on the other side of the plane P. v fv (P) Fig. 1 Reflection. The concept of reflection is defined by three equations: The reflection plane P is determined by a vector v of length 1 (3.1) llvll = 1 Reflection preserves the norm of a vector (3.2) llall = llbll The difference between a vector a and its reflection b is a vector fv which is a multiple of v (3.3) fv - a- b HOUSEHOLDER REDUCTION 5 The (unknown) scalar f is the distance between the vector and its reflection. We must find the reflection of an arbitrary vector a through a plane P defined by a given unit vector v. by (3.2) (a- fv)T(a- fv) by (2.4), (3.3) by (2.2), (2.4), (3.1) This equality determines the distance f between vector a and its image b (3.4) The reflection of b into a displaces b by the same distance f in the opposite direction. So we can also express the distance as (3.5) Finally we define b in terms of a and v b =a- vf by (3.3) by (3.4) where I is then x n identity matrix defined in the appendix. In other words, the reflection of a vector a is the vector (3.6) b = Ha obtained by multiplying a by then x n reflection matrix (3.7) H = I- 2vvT His also called a Householder matrix. This is the rabbit that is often pulled out of the hat without any explanation of why it has this particular form. Figure 1 is a geometric definition of reflection in 3-dimensional space. However, the algebraic equations derived from this figure make no assumptions about the dimension of space. In the following, we will simply say that Eqs. (3.6) and (3. 7) define a transformation of ann-dimensional vector. By analogy we will call this transformation a "reflection" through an (n- I)-dimensional plane. The essential property is that reflection of an n-dimensional vector preserves the norm (3.8) IIHall = llall HOUSEHOLDER REDUCTION 6 This follows from Eqs. (3.2) and (3.6). If we reflect a vector twice through the same plane, we get the same vector again H(Ha) = a In other words, two reflections are equivalent to an identity transformation HH =I Consequently H is a non-singular matrix which is its own inverse s-1 = H (see the appendix). 4. Householder reduction. We are looking for an algorithm that reduces an n x n real matrix A to triangular form without increasing the magnitude of the elements significantly. An element of a column can never exceed the total length of the column vector. That is iaiil ~ lladl for i,j = 1, 2, ... , n In other words, the norm of a column vector is an upper bound on the magnitude of its elements. A method that changes the elements of a matrix A without changing the norms of its columns will obviously limit the magnitude of the matrix elements. This can be achieved by multiplying A by a Householder matrix H. If we multiply a system of linear equations Ax = b by a non-singular matrix H, we obtain an equation (HA)x = Hb that has the same solution as the original system. The first step in Householder reduction produces a matrix H A that has all zeros below the first element of the first column. The reflection must transform column (4.1) into a column of the form (4.2) (du 0 ... O]T HOUSEHOLDER REDUCTION 7 where the diagonal element is (4.3) The choice of sign will be made later. Equations ( 4.1 )-( 4.3) define the computation of the first column of the matrix HA. The difference between column a1 and its reflection H a1 is the column vector by (3.3) by (3.6) Combining this with Eqs. (4.1) and (4.2) we find (4.4) where the first element is (4.5) Wn = an - dn The distance between a1 and its image H a1 is f 1 where In short (4.6) fi(-2vTHal) - -2(fivfHal = -2wn dn by (3.5), (3.6) by (2.1), (4.2), (4.4) The unit vector v which determines the appropriate Householder matrix is or by Eq. ( 4.4) (4.7) After the transformation of the first column a1, each remaining column ai is also replaced by its reflection through the same plane defined by Eqs. (3.3), (3.4), and (3.6). (4.8) HOUSEHOLDER REDUCTION 8 (4.9) fi = 2vT ai The reflection of a column is obtained by subtracting a multiple of the unit vector v. 5. Numerical stability. vVe still need to decide which sign to use for the diagonal element du in Eq. (4.3). If du = au, the scalars wu and /1 are zero by Eqs. (4.5) and (4.6), and the division by / 1 in Eq. (4.7) causes overflow. We can avoid this problem by selecting the sign which makes dn =f=. an. The overflow problem occurs when a1 is a multiple of the unit vector el = [1 0 ... of For a = au e1 there are four cases to consider au> 0: au< 0: du = + lla1ll = au (overflow) du = - lla1ll = -an (no overflow) du = + lla1ll = dn =- lla1ll = -au au (no overflow) (overflow) If a1 is close to a multiple of et, serious rounding errors may occur if / 1 is very small. This insight leads to the following rule (5.1) 6. Computational rules. We are now ready to summarize the rules for com-puting the matrix HAas defined by Eqs. (2.1), (4.2), (4.5)-(4.9), and (5.1): lla1ll dn wn (6.1) /I Ha1 v /i Hai #. if au > 0 then -lla111 else lla1ll au- du ..j-2wudu [du 0 ... of - [wu a21 ... anl]T //1 2vT ai for 1 < i ~ n - ai- fiv Householder's algorithm reduces a system of linear equations to upper triangular form in n - 1 steps: The first step reduces A to a matrix H A with all zeros below the diagonal element in the first column. At the same time, b is transformed into a vector Hb. This computation, defined by Eq. (6.1), is called a Householder transformation. HOUSEHOLDER REDUCTION 9 HA Hb * * ... * * 0 I i ~ I * 0 * 0 * The second step reduces the (n- 1) X (n -1) submatrix of HA shown above by Householder transformation. We now obtain a matrix with zeros below the diagonal elements in the first two columns. The same transformation is applied to the ( n-1) x 1 subvector of Hb shown above. By a series of Householder transformations, applied to smaller and smaller sub-matrices and subvectors, the equation system is reduced, one column at a time, to upper triangular form. 7. A numerical example. We now return to the previous example of three equations with three unknowns. For convenience we combine the matrix A and the vector b into a single 3 x 4 matrix [ 2 2 4 18] AO = 1 3 -2 1 3 1 3 14 First we reduce AO to a matrix A1 with all zeros below the diagonal element in the first column. This is done column by column using Eq. (6.1). The numbers shown below were produced by a computer and rounded to four decimal places. First column: al (2 1 3jT v [0.8759 0.1526 0.4577JT fl 6.5549 Ha1 [-3.7417 o o]r Second column: a2 (2 3 1]T h 5.3344 Ha2 -[-2.6726 2.1862 -1.4414]T Third column: a3 (4 -2 3)T h - 9.1433 Ha3 - [-4.0089 -3.3949 -1.1846)T Fourth column: a4 - (18 1 14)T !4 - 44.6536 Ha4 - [-21.1136 -5.8123 -6.4368jT to HOUSEHOLDER REDUCTION We now have the matrix [ -3.7417 Al = 0 0 -2.6726 -4.0089 -21.1136] 2.1862 -3.3949 -5.8123 -1.4414 -1.1846 -6.4368 The next step of the algorithm reduces the 2 x 2 submatrix Al' = [ 2.1862 -3.3949 -5.8123] -1.4414 -1.1846 -6.4368 A2, = [ -20.6186 2.1822 1.3093] -2.8577 -8.5732 The final triangular matrix [ -3.7417 A2 = 0 0 -2.6726 -4.0089 -21.1136] -2.6186 2.1822 1.3093 0 -2.8577 -8.5732 consists of the first row and column of AI and the submatrix A2'. The triangular equation system is solved by back substitution to obtain x = [Loooo 2.oooo 3.oooof 10 8. Pascal algorithm. The following Pascal algorithm assumes that the matrix A is stored by columns, that is, a[i) denotes the ith column of A. For each submatrix of A, the eliminate operation is applied to the first column, and the transform operation is applied to each remaining column (including b). type column = array [l..n) of real; matrix= array [l..n] of column; procedure reduce (var a: matrix; var b: column); var vi: column; i, j: integer; function product(i: integer; var a, b: column): real; { the scalar product of elements i .. n of a and b } var ab: real; k: integer; begin ab := 0.0; fork:= ito n do ab := ab + a[k]*b[k]; product := ab end; HOUSEHOLDER REDUCTION procedure eliminate(i: integer; var ai, vi: column); var anorm, dii, fi, wii: real; k: integer; begin anorm := sqrt(product(i, ai, ai)); if ai[i] > 0.0 then dii := -anorm else dii := anorm; wii := ai[i] - dii; fi := sqrt( -2.0*wii*dii); vi[i] := wii/fi; ai[i] := dii; for k := i + 1 to n do begin vi[k] := ai[k] /fi ; ai[k] := 0.0 end end; procedure transform(i: integer; var aj, vi: column); var fi: real; k: integer; begin fi := 2.0*product(i, vi, aj); for k := i to n do end; begin aj[k] := aj[k] - fi*vi[k] for i := 1 to n - 1 do begin eliminate (i, a[i], vi); for j := i + 1 to n do transform(i, a[j], vi); transform(i, b, vi); end end The execution time of the algorithm is O(n3 ). 11 Final remarks. We have explained Householder's method for reducing a matrix to triangular form. The main advantage of the method is that it achieves numerical stability without pivoting. We have illustrated the computation by an example and a HOUSEHOLDER REDUCTION 12 Pascal algorithm. Householder reduction is an interesting example of a fundamental computation with a subtle theory and a short algorithm. Acknowledgements. To Fred Schlereth for bringing Householder's method to my attention, Nawal Copty for explaining it to me, and to Jonathan Greenfield and Erik Hemmingsen for advice on how to improve the presentation. REFERENCES E. KREYSZIG [1988], Advanced Engineering Mathematics, John Wiley & Sons, New York. A. S. HOUSEHOLDER [1958], "Unitary triangularization of a nonsymmetricmatrix," J. ACM, 5, pp. 339-342. W. H. PRESS, B. P. FLANNERY, S. A. TEUKOLSKY, and W. T. VETTERLING [1989], Numerical Recipes in Pascal-The Art of Scientific Computing, Cambridge Uni-versity Press, New York. J. M. ORTEGA [1988], Introduction to Parallel and Vector Solution of Linear Sys-tems, Plenum Press, New York. P. BRINCH HANSEN [1990], The All-Pairs Pipeline, School of Computer and Infor-mation Science, Syracuse University, Syracuse, New York, 1990. G. C. FOX, M. A. JOHNSON, G. A. LYZENGA, S. W. OTTO, J. K. SALMON, and D. W. WALKER [1988], Solving Problems on Concurrent Processors, Vol. I, Prentice-Hall, Englewood Cliffs, New Jersey. APPENDIX: MATRIX ALGEBRA In the algebraic laws, A, B, and C denote matrices, while k is a scalar. The identity matrix is 1 0 0 0 0 1 0 0 I= 0 0 1 0 0 0 0 1 The transpose AT is the matrix obtained by exchanging the rows and columns of the matrix A. HOUSEHOLDER REDUCTION The inverse of a matrix A is a matrix A -l such that If A -I exists then A is called a non-singular matrix. The laws also apply to vectors since they are n X 1 (or 1 x n) matrices. Identity Law: Symmetry Law: Associative Laws: Distributive Laws: Transposition Laws: Scaling Laws: IA AI = A A+B A ± (B ± C) A(BC) A(B ± C) (A ± B)C (AT)T (A± B)Y (AB)T B+A (A ± B) ± C (AB)C AB ± AC AC ± BC I A kA = Ak k(AB) = (kA)B = A(kB) kAT = (kA? 13 

HOUSEHOLDER REDUCTION

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HOUSEHOLDER REDUCTION

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