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Duality relations associated with Gaussian elimination
U.S.S.R. Comput.Maths.Math.Phys.Vol.23,No.l,pp.149-151,1983 Printed in Great Britain
OO41-5553/83 $iO.00+O.OO 9 1984 Pergamon Press Ltd.
S H O R T COMMUNICATIONS
DUALITY RELATIONS ASSOCIATED WITH GAUSSIAN ELIMINATION*
Kh.D. IKRAMOV Using the duality relations associated w i t h Gauss's method it is shown that: i. Any main submatrix of a 'totally non-degenerate matrix A is stipulated (in the sense of an arbitrary monotonic norm) to be not inferior to A itself; 2. If the inverse of matrix A is diagonally dominant with respect to the columns, then the linear system A x = b is regularly solved by Jordan's method without the main element being chosen. i. Throughout, A denotes a square non-degenerate matrix of order n with non-zero corner minors, and B a n inverse of A. We distinguish between the concept of a m a i n submatrix, that is of the arbitrary submatrix located on the intersection of groups of rows and columns with the same numbers, a n d the corner submatrix as a main submatrix located at either the left upper or the right lower corner of a given matrix (and correspondingly, we speak of u p p e r a n d lower corner submatrices). The same applies to the concept of the main and corner minors. We denote by A~(AkI) the upper (lower) corner submatrix of matrix A, of order k. ~W,, DDR., DDC,, TN~', DDRn -~, DDC,-, denote, in the order indicated, the classes nXn of the nondegenerate totally non-negative matrices, the matrices diagonally predominant with respect to rows, matrices diagonally predominant with respect to columns, and the corresponding classes of the inverse matrices. We note that the matrix norm Ib,!i is referred to as monotonic if for the arbitrary matrices F and G with identical dimensions from l/~JI~lg0l we have iI~[l~ilG~l for any i,].In particular, the well-known norms Iblh,If.If.,If'TiE and the norm IEa[l~=n max
are monotonic.
I~ol.
~'J
2. The majority of the statements of this paragraph are based on the relation formulated in Lemma i, between the main submatrices of A, and the submatrices of B, obtained by Gaussian elimination (the parts played by A and B a r e of course interchangeable). Assume that the order of rows and columns o f A , and therefore of B, is fixed.We denote b y B (h' the lower corner n-th order submatrix of the matrix B,_k=(b~~) obtained from B as a result of n--k steps of the Gaussian method. L e m m a i. The relation B(h)=(A~l)-' holds. This is a consequence of the Frobenius formula and is noted, for example, in /1/. C o n s i d e r i n g t h e arbitrary symmetrical re-ordering of the rows and columns of A , we obtain a correspondence between all m a i n submatrices of A, and the submatrices which are potentially contained in B and which can be exposed by Gaussian exclusion. An immediate consequence of Lenaaa 1 is the possibility of interpreting the growth of the elements in the Gaussian method for the matrix B as a relative conditionality of the corner submatrices of the matrix A. Namely, we assume that
b~= max ~b~-kl. LJ>n-~ Lemma
2.
The relation
ll(A~)-'llH=kb h holds.
Theorem i. Let H be one of the symbols TN, DDN, DDC, TN-',DDB -t, DDC-'. i. The arbitrary main n-th order submatrix A~ of matrix A belongs to
2.
If A~H., then: Hh;
A(A)~IIh.
P r o o f . The first statement is obvious for H = { T ~ DDR, DDC} and by Lenaaa 1 it involves the second statement for H={TN-', DDR -i,DDC-'}. It is also w e l l - k n o w n t h a t t h e property of the diagonal dominance (in respect to b o t h rows and columns) is complete in the Gaussian m e t h o d in the sense that it is transmitted from A to the submatrices A (h) (see, for example, /i/, /2/); this is exactly Statement 2 for H={DDR, DDC), and hence follows Statement 1 for H={DDR-,, DDC-~). It remains to prove Statement 2 for the case H = T ~ We shall refer to the result obtained in /3/: if AeTN., a lower triangular matrix L and an upper triangular matrix R exist, such t h a t A=LR and L, B~TN.. Clearly, the submatrices LhI and R, 1 produce the triangular factorization of the matrixA(k), hence A(h)~L%i. This results in Statement 1 for the case H=TN-L -Lenm~ 2 and the estimates of growth of the elements in the Gaussian m e t h o d for the classes of matrices discussed enables us to estimate the conditionality of their main submatrices. Theorem 2. i. If AEH,, and H={TN, TN-'} then a. bh~bo, c0nd(A). Here II.lI is the arbitrary monotonic matrix form. *Zh.vychisl. Mat.mat. Fiz., Vol.23,No.l,213-216,1983 149
b.
[IAk-tII~[[A-'I[, c. c0nd(Ah)~
150
2. I f 2cond.,~(.4).
A~H., II={DDB, DDc, DDR-t, DDC-t}, t-hen
a.
b~2bo,
b . IIAh-'lIM~21tA-'Jl.v, c .
condM(Ah)<
Proof. It is shown in /i/ that if B~TN, or B~TN, -i, then the relations Ib~-~" [~]b0[, i, I> n-k hold for the similarly-named elements of matrices B and B,-k (B,-~, is obtained from B by means o f - n - k steps in the Gaussian method). These inequalities overlap the relation *la of the theorem and in the obvious way they entail validity of ib and ic. The fact that the growth of elements in the Gaussian method, i.e., b~,[bo is limited to 2 in the case of a diagonally predominant B is well known (see /i/ and /2/). Therefore it remains only to show that condition 2a is satisfied for II={DDB, DDC}. This will be obvious from the consequence of Lemma 3, which is also of interest itself. Lemma
3.
Let the inequality 1
IdetAl<: a[det A~U[ IdetA._h] .
(1)
be valid for the pair of complementary corner submatrices Ahu andAl_~, and for the positive number a. Then the analogous inequality
[detB[~; a[detB,~ [detB~-k[. holds for the inverse matrix
(2)
B = A -t.
Proof.
Inequality (2) can be obtained if both parts of (i) are divided by (detA)Z , and the well-known relation between mutually reciprocal matrices A and B is used (see /4/). Now consider the proof of Theorem 2. We wish to show that in using the Gaussian method for matrix B from class DDR, -t or DDCn -~ the growth of the elements does not exceed 2. The inverse of the diagonally predominant matrix with respect to rows (or columns) has the following property: in each column (row) the element with the highest mdoulus is on the principal diagonal /5/. By Theorem i, this property is extended to the submatrix B(hL We assume that the maximum element of B(k~with respect to modulus is at position (i.i), which can always be achieved by a symmetric transposition of B. Assuming p=n-k+i, and considering that (b(Jkt)= n-& b;~ ), we prove the inequality
bs=lb,(,~l< 2lbprl < 2bo.
(3)
In fact, the inequality
b,',~'= (,tot BU+,) (d~t 8;r holds for the element
#~) 9 vii
Thus, inequality
(3) is equivalent to the inequality
Idet B u d < 21b~l Idet Bpul.
(4)
By L e n a 3, inequalitYu(4) holds if an analogous inequality exists for an inverse of BvU.,. However, the matrix (Bp~t)-i is diagonally predominant, and inequality (4) was proved in /5/ for this class of matrices. 3.
The factorize,ion A = LDR,
(5 )
is understood as a triangular factorization of the matrix A where L and B are triangle matrices (the upper and lower respectively) with unit diagonal elements, and D is a diagonal matrix. By the assumptions made in paragraph 1 with respect to A, the trihngular factorization (5) exists and is unique. We introduce additionally the notation I:=LD and I~=DR.
Theorem 3. 3. If
A~DDB,-'
i. If Ar then BeDDR,, I{EDDB.. 2. If A e D D C . then L~DDC,,, E~DDC.. then L ~ D D R , -t, E~DDH,-t 4. If A ~ DDC,-' then B~DDC.-t I ~ D D C ~ - L
Proof. Clearly, Statements 2 and 4 are obtained by applying Statements 1 and 3 to the transpose of the matrices. Statement 1 is well known /i/, and therefore it remains to prove Statement 3. Let P be a matrix of transposition and have the form
We assume that [~=PBP. the centre. If AEDDR,-', then ization of
This matrix is obtained from B by the reflection of B with respect to B=A-'~DDR,,
this being true for B as well.
I3=LtDtBt
In the triangular factor(6)
and of B,, I~t~DDB~. By Eq. (6) , B=L,3,Bt and B-t=A=Bt-tD~-tL, -t. By comparing this equation with Eq. (5), we can say that the matrices L and E are the inverse of the matrices It, and D j G = Bi 9 that is L, EcDDB~-,. Now, let the linear algebraic system A x = b be solvable by Jordan's method without choosing a main element. Theorem
4.
If
A~DDC.-'
then Jordan's method is stable.
151
Proof. By the analysis given in /6/, it is sufficient to show that the following is true fo the matrix A ~ D D C . -t : i. In a straightforward operation of Gauss's method, a~2=o; 2. There is no growth of elements in the intermediate matrices of Jordan's method in solving a triangular system with the matrix R Statement 1 is included in Statement 2a of Theorem 2. AS regards Statement 2 we note that H ~ D D H . -t (see item 4 of Theorem 3). Now Statement 1 follows from the next lemma. L e m m a 4. Let R be an upper triangular matrix with unit diagonal elements, and R~=(ro*) a matrix obtained from R as a result of k steps of the Jordan elimination (Ho=R. R.=l,; ;. being a unit matrix of order n). Then, if R~DDR,~-, or R~DDC,-,. we have Ir~I~l for any I,]. and k.
Proof. We shall show that I~eDDR,-,(DDC,-') follows from REDDR,-'(DDC,-t) for all k. Thus, the element of matrix H~ with the highest modulus is on the principal diagonal and equals L The matrices R -i and Bh-~, when divided into blocks, have the form
Here Zh and A.-~ are triangular submatrices with unit elements, and the subscripts indicate the dimensions of the blocks. Hence it is clear that Rh-t retains the same form of diagonal predominance as did R-'. REFERENCES i. FURSANOV E.P., On the growth of elements in Gauss elimination, in: Numerical analysis with FORTRAN (Chislennyi analiz na FORTRANE) Series XIV, MGU, Moscow, 48-67, 1976. 2. STEWART G.W., Introduction to matrix computations. Academic Press, New York, 1973. 3. CRYER C.W., The LU-factorization of totally positive matrices, Linear Algebra and Applications, 7, l, 83-92, 1973. 4. GANTMAKHER F.P., Theory of matrices, (Teoriya matrits), Nauka, Moscow, 1966. 5. IKRAMOV Kh.D., On the conditionality of the intermediate matrices of the Gauss and Jordan methods, and of the optimal exclusion, Zh.vychisl.Mat.mat.Fiz., 18, 3, 531-545, 1978. 6. IKRAMOV Kh.D., On an approach to the analysis of the rounding error of non-orthogonal methods for solving systems of linear algebraic equations, Zh.vychlsl. Mat.mat.Fiz., 22, 2, 259-268, 1982. Translated by W.C.
OO41-5553/83 $iO.00+O.OO 9 1984 Pergamon Press Ltd.
S H O R T COMMUNICATIONS
DUALITY RELATIONS ASSOCIATED WITH GAUSSIAN ELIMINATION*
Kh.D. IKRAMOV Using the duality relations associated w i t h Gauss's method it is shown that: i. Any main submatrix of a 'totally non-degenerate matrix A is stipulated (in the sense of an arbitrary monotonic norm) to be not inferior to A itself; 2. If the inverse of matrix A is diagonally dominant with respect to the columns, then the linear system A x = b is regularly solved by Jordan's method without the main element being chosen. i. Throughout, A denotes a square non-degenerate matrix of order n with non-zero corner minors, and B a n inverse of A. We distinguish between the concept of a m a i n submatrix, that is of the arbitrary submatrix located on the intersection of groups of rows and columns with the same numbers, a n d the corner submatrix as a main submatrix located at either the left upper or the right lower corner of a given matrix (and correspondingly, we speak of u p p e r a n d lower corner submatrices). The same applies to the concept of the main and corner minors. We denote by A~(AkI) the upper (lower) corner submatrix of matrix A, of order k. ~W,, DDR., DDC,, TN~', DDRn -~, DDC,-, denote, in the order indicated, the classes nXn of the nondegenerate totally non-negative matrices, the matrices diagonally predominant with respect to rows, matrices diagonally predominant with respect to columns, and the corresponding classes of the inverse matrices. We note that the matrix norm Ib,!i is referred to as monotonic if for the arbitrary matrices F and G with identical dimensions from l/~JI~lg0l we have iI~[l~ilG~l for any i,].In particular, the well-known norms Iblh,If.If.,If'TiE and the norm IEa[l~=n max
are monotonic.
I~ol.
~'J
2. The majority of the statements of this paragraph are based on the relation formulated in Lemma i, between the main submatrices of A, and the submatrices of B, obtained by Gaussian elimination (the parts played by A and B a r e of course interchangeable). Assume that the order of rows and columns o f A , and therefore of B, is fixed.We denote b y B (h' the lower corner n-th order submatrix of the matrix B,_k=(b~~) obtained from B as a result of n--k steps of the Gaussian method. L e m m a i. The relation B(h)=(A~l)-' holds. This is a consequence of the Frobenius formula and is noted, for example, in /1/. C o n s i d e r i n g t h e arbitrary symmetrical re-ordering of the rows and columns of A , we obtain a correspondence between all m a i n submatrices of A, and the submatrices which are potentially contained in B and which can be exposed by Gaussian exclusion. An immediate consequence of Lenaaa 1 is the possibility of interpreting the growth of the elements in the Gaussian method for the matrix B as a relative conditionality of the corner submatrices of the matrix A. Namely, we assume that
b~= max ~b~-kl. LJ>n-~ Lemma
2.
The relation
ll(A~)-'llH=kb h holds.
Theorem i. Let H be one of the symbols TN, DDN, DDC, TN-',DDB -t, DDC-'. i. The arbitrary main n-th order submatrix A~ of matrix A belongs to
2.
If A~H., then: Hh;
A(A)~IIh.
P r o o f . The first statement is obvious for H = { T ~ DDR, DDC} and by Lenaaa 1 it involves the second statement for H={TN-', DDR -i,DDC-'}. It is also w e l l - k n o w n t h a t t h e property of the diagonal dominance (in respect to b o t h rows and columns) is complete in the Gaussian m e t h o d in the sense that it is transmitted from A to the submatrices A (h) (see, for example, /i/, /2/); this is exactly Statement 2 for H={DDR, DDC), and hence follows Statement 1 for H={DDR-,, DDC-~). It remains to prove Statement 2 for the case H = T ~ We shall refer to the result obtained in /3/: if AeTN., a lower triangular matrix L and an upper triangular matrix R exist, such t h a t A=LR and L, B~TN.. Clearly, the submatrices LhI and R, 1 produce the triangular factorization of the matrixA(k), hence A(h)~L%i. This results in Statement 1 for the case H=TN-L -Lenm~ 2 and the estimates of growth of the elements in the Gaussian m e t h o d for the classes of matrices discussed enables us to estimate the conditionality of their main submatrices. Theorem 2. i. If AEH,, and H={TN, TN-'} then a. bh~bo, c0nd(A). Here II.lI is the arbitrary monotonic matrix form. *Zh.vychisl. Mat.mat. Fiz., Vol.23,No.l,213-216,1983 149
b.
[IAk-tII~[[A-'I[, c. c0nd(Ah)~
150
2. I f 2cond.,~(.4).
A~H., II={DDB, DDc, DDR-t, DDC-t}, t-hen
a.
b~2bo,
b . IIAh-'lIM~21tA-'Jl.v, c .
condM(Ah)<
Proof. It is shown in /i/ that if B~TN, or B~TN, -i, then the relations Ib~-~" [~]b0[, i, I> n-k hold for the similarly-named elements of matrices B and B,-k (B,-~, is obtained from B by means o f - n - k steps in the Gaussian method). These inequalities overlap the relation *la of the theorem and in the obvious way they entail validity of ib and ic. The fact that the growth of elements in the Gaussian method, i.e., b~,[bo is limited to 2 in the case of a diagonally predominant B is well known (see /i/ and /2/). Therefore it remains only to show that condition 2a is satisfied for II={DDB, DDC}. This will be obvious from the consequence of Lemma 3, which is also of interest itself. Lemma
3.
Let the inequality 1
IdetAl<: a[det A~U[ IdetA._h] .
(1)
be valid for the pair of complementary corner submatrices Ahu andAl_~, and for the positive number a. Then the analogous inequality
[detB[~; a[detB,~ [detB~-k[. holds for the inverse matrix
(2)
B = A -t.
Proof.
Inequality (2) can be obtained if both parts of (i) are divided by (detA)Z , and the well-known relation between mutually reciprocal matrices A and B is used (see /4/). Now consider the proof of Theorem 2. We wish to show that in using the Gaussian method for matrix B from class DDR, -t or DDCn -~ the growth of the elements does not exceed 2. The inverse of the diagonally predominant matrix with respect to rows (or columns) has the following property: in each column (row) the element with the highest mdoulus is on the principal diagonal /5/. By Theorem i, this property is extended to the submatrix B(hL We assume that the maximum element of B(k~with respect to modulus is at position (i.i), which can always be achieved by a symmetric transposition of B. Assuming p=n-k+i, and considering that (b(Jkt)= n-& b;~ ), we prove the inequality
bs=lb,(,~l< 2lbprl < 2bo.
(3)
In fact, the inequality
b,',~'= (,tot BU+,) (d~t 8;r holds for the element
#~) 9 vii
Thus, inequality
(3) is equivalent to the inequality
Idet B u d < 21b~l Idet Bpul.
(4)
By L e n a 3, inequalitYu(4) holds if an analogous inequality exists for an inverse of BvU.,. However, the matrix (Bp~t)-i is diagonally predominant, and inequality (4) was proved in /5/ for this class of matrices. 3.
The factorize,ion A = LDR,
(5 )
is understood as a triangular factorization of the matrix A where L and B are triangle matrices (the upper and lower respectively) with unit diagonal elements, and D is a diagonal matrix. By the assumptions made in paragraph 1 with respect to A, the trihngular factorization (5) exists and is unique. We introduce additionally the notation I:=LD and I~=DR.
Theorem 3. 3. If
A~DDB,-'
i. If Ar then BeDDR,, I{EDDB.. 2. If A e D D C . then L~DDC,,, E~DDC.. then L ~ D D R , -t, E~DDH,-t 4. If A ~ DDC,-' then B~DDC.-t I ~ D D C ~ - L
Proof. Clearly, Statements 2 and 4 are obtained by applying Statements 1 and 3 to the transpose of the matrices. Statement 1 is well known /i/, and therefore it remains to prove Statement 3. Let P be a matrix of transposition and have the form
We assume that [~=PBP. the centre. If AEDDR,-', then ization of
This matrix is obtained from B by the reflection of B with respect to B=A-'~DDR,,
this being true for B as well.
I3=LtDtBt
In the triangular factor(6)
and of B,, I~t~DDB~. By Eq. (6) , B=L,3,Bt and B-t=A=Bt-tD~-tL, -t. By comparing this equation with Eq. (5), we can say that the matrices L and E are the inverse of the matrices It, and D j G = Bi 9 that is L, EcDDB~-,. Now, let the linear algebraic system A x = b be solvable by Jordan's method without choosing a main element. Theorem
4.
If
A~DDC.-'
then Jordan's method is stable.
151
Proof. By the analysis given in /6/, it is sufficient to show that the following is true fo the matrix A ~ D D C . -t : i. In a straightforward operation of Gauss's method, a~2=o; 2. There is no growth of elements in the intermediate matrices of Jordan's method in solving a triangular system with the matrix R Statement 1 is included in Statement 2a of Theorem 2. AS regards Statement 2 we note that H ~ D D H . -t (see item 4 of Theorem 3). Now Statement 1 follows from the next lemma. L e m m a 4. Let R be an upper triangular matrix with unit diagonal elements, and R~=(ro*) a matrix obtained from R as a result of k steps of the Jordan elimination (Ho=R. R.=l,; ;. being a unit matrix of order n). Then, if R~DDR,~-, or R~DDC,-,. we have Ir~I~l for any I,]. and k.
Proof. We shall show that I~eDDR,-,(DDC,-') follows from REDDR,-'(DDC,-t) for all k. Thus, the element of matrix H~ with the highest modulus is on the principal diagonal and equals L The matrices R -i and Bh-~, when divided into blocks, have the form
Here Zh and A.-~ are triangular submatrices with unit elements, and the subscripts indicate the dimensions of the blocks. Hence it is clear that Rh-t retains the same form of diagonal predominance as did R-'. REFERENCES i. FURSANOV E.P., On the growth of elements in Gauss elimination, in: Numerical analysis with FORTRAN (Chislennyi analiz na FORTRANE) Series XIV, MGU, Moscow, 48-67, 1976. 2. STEWART G.W., Introduction to matrix computations. Academic Press, New York, 1973. 3. CRYER C.W., The LU-factorization of totally positive matrices, Linear Algebra and Applications, 7, l, 83-92, 1973. 4. GANTMAKHER F.P., Theory of matrices, (Teoriya matrits), Nauka, Moscow, 1966. 5. IKRAMOV Kh.D., On the conditionality of the intermediate matrices of the Gauss and Jordan methods, and of the optimal exclusion, Zh.vychisl.Mat.mat.Fiz., 18, 3, 531-545, 1978. 6. IKRAMOV Kh.D., On an approach to the analysis of the rounding error of non-orthogonal methods for solving systems of linear algebraic equations, Zh.vychlsl. Mat.mat.Fiz., 22, 2, 259-268, 1982. Translated by W.C.