fast and accurate computations with totally nonnegative ... · fast and accurate computations with...

Fast and accurate computations with TotallyNonnegative Quasiseparable Matrices

Froilán M. Dopico1

Tom Bella2 Vadim Olshevsky3

1Departamento de Matemáticas, Universidad Carlos III de Madrid

2Department of Mathematics, University of Rhode Island

3Department of Mathematics, University of Connecticut

Positive Systems: Theory and Applications, September 2-4, 2009,Valencia

F. M. Dopico (U. Carlos III, Madrid) Totally Nonnegative Quasiseparable Matrices POSTA 09 1 / 45

Outline

1 Quasiseparable matrices

2 Goals of the talk

3 Neville elimination and quasiseparable matrices

4 Totally Nonnegative (TN) quasiseparable matrices

5 Solving quasiseparable linear systems

6 Error analysis for quasiseparable linear systems

7 Conclusions and future work

Outline

2 Goals of the talk

Quasiseparable matrices (I): Definition

DefinitionA square matrix C is quasiseparable of order (nL, nU ) if

every submatrix of C entirely located in the strictly lower (resp.upper) triangular part of C have rank at most nL (resp. nU ),andat least one of these submatrices has rank equal to nL (resp. nU ).

RemarkIn this talk, we are interested only in the order (1,1) and for brevity thesimple term quasiseparable is used instead of (1, 1)-quasiseparable.

It is necessary and sufficient that the following submatrices have rankat most 1:

Quasiseparable matrices (II)

× × × × ×× × × × ×× × × × ×× × × × ×× × × × ×

Green’s quasiseparable matrices (I)

Definition (Green’s quasiseparable matrices)A square matrix G is Green’s quasiseparable of order (1, 1) if

every submatrix of G entirely located in the lower (resp. upper)triangular part (including the diagonal) of G have rank at most1 (resp. 1), andat least one of these submatrices has rank equal to 1 (resp. 1).

Green’s quasiseparable matrices (I)

Definition (Green’s quasiseparable matrices)A square matrix G is Green’s quasiseparable of order (1, 1) if

every submatrix of G entirely located in the lower (resp. upper)triangular part (including the diagonal) of G have rank at most1 (resp. 1), andat least one of these submatrices has rank equal to 1 (resp. 1).

Green’s quasiseparable matrices (II)

× × × × ×× × × × ×× × × × ×× × × × ×× × × × ×

Parametrization of quasiseparable matrices

Theorem (Eidelman and Gohberg (1999))The set of n× n quasiseparable matrices can be parameterized interms of 7n− 8 independent parameters or generators.

Example (Every 5× 5 quasiseparable matrix is of the form)

d1 g1h2 g1b2h3 g1b2b3h4 g1b2b3b4h5

p2q1 d2 g2h3 g2b3h4 g2b3b4h5

p3a2q1 p3q2 d3 g3h4 g3b4h5

p4a3a2q1 p4a3q2 p4q3 d4 g4h5

p5a4a3a2q1 p5a4a3q2 p5a4q3 p5q4 d5

RemarkThere are seven vectors (families) of parameters: p2:n, a2:n−1, q1:n−1,d1:n, g1:n−1, b2:n−1, h2:n.

Parametrization of Green’s quasiseparable

TheoremThe set of n× n Green’s quasiseparable matrices can be parameterized interms of 6n− 2 parameters with the constraints piqi = gihi for i = 1 : n.

Example (Every 5× 5 Green’s quasiseparable matrix is of theform)

p1q1 g1b1h2 g1b1b2h3 g1b1b2b3h4 g1b1b2b3b4h5

p2a1q1 p2q2 g2b2h3 g2b2b3h4 g2b2b3b4h5

p3a2a1q1 p3a2q2 p3q3 g3b3h4 g3b3b4h5

p4a3a2a1q1 p4a3a2q2 p4a3q3 p4q4 g4b4h5

p5a4a3a2a1q1 p5a4a3a2q2 p5a4a3q3 p5a4q4 p5q5

RemarkThere are six vectors of parameters: p1:n, a1:n−1, q1:n, g1:n, b1:n−1, h1:n.

Highlights on research on quasiseparable matrices (I)

During the last six years an intense research has been performed onquasiseparable matrices.

Many researchers from different countries: Belgium, Check Republic,Italy, Israel, Russia, USA...

Quasiseparable matrices appear in many applications: systems theoryand signal processing, discretization of integral equations, covariancematrices in multivariate statistics, discretization of elliptic PDEs....

Quasiseparable matrices include many important classes ofmatrices: companion matrices of polynomials, tridiagonal matricesand their inverses (Green’s quasiseparable), unitary Hessenberg,banded matrices (for order larger than (1, 1)),.....

Inverses of quasiseparable matrices are quasiseparable.

Highlights on research on quasiseparable matrices (II)

A main line of research has been the development of structured fastalgorithms by using the low number of parameters defining thisclass. There are many algorithms an their costs are:

Problem Cost of traditional Cost of structuredalgorithms quasiseparable algs.

systems of equations O(n3) O(n)eigenvalues O(n3) O(n2)singular values O(n3) O(n2)

The stability of these algorithms is not guaranteed and, as far as weknow, error analysis have not been developed even for the mostsimple cases.

Highlights on research on quasiseparable matrices (II)

A main line of research has been the development of structured fastalgorithms by using the low number of parameters defining thisclass. There are many algorithms an their costs are:

Problem Cost of traditional Cost of structuredalgorithms quasiseparable algs.

systems of equations O(n3) O(n)eigenvalues O(n3) O(n2)singular values O(n3) O(n2)

The stability of these algorithms is not guaranteed and, as far as weknow, error analysis have not been developed even for the mostsimple cases.

An historical TN-quasiseparable connection

In Oscillation Matrices (1941) by Gantmacher and Krein a particular classof symmetric Green’s quasiseparable matrices is considered. These arecalled single-pair matrices and are defined as

p1q1 q1p2 q1p3 . . .p2q1 p2q2 q2p3 . . .p3q1 p3q2 p3q3 . . .

......

.... . .

= tril(pqT ) + strict-triu(qpT ),

where all the numbers p = [p1, . . . , pn]T , q = [q1, . . . , qn]T are nonzero.

These matrices are obtained from general Green’s quasiseparablematrices by taking ai = bi = 1, gi = qi, and hi = pi.

Theorem (Gantmacher and Krein (1941))

S is TN if and only if all the numbers p1, . . . , pn, q1, . . . , qn have the same signand

q1p1≤ q2p2≤ · · · ≤ qn

......

.... . .

q1p1≤ q2p2≤ · · · ≤ qn

......

.... . .

q1p1≤ q2p2≤ · · · ≤ qn

Outline

2 Goals of the talk

Goals of the talk

We initiate the study of stability of fast algorithms for quasiseparablematrices, by presenting rounding errors analysis of the solution ofquasiseparable linear systems by using a bidiagonal factorizationfollowed by a Björck-Pereyra type algorithm.

We prove that this algorithm is componentwise backward stable in astrong sense, i.e., preserving the structure, in the TN-quasiseparablecase.

For Green’s quasiseparable matrices simple forward errors boundsfor this algorithm are presented and we show that it is frequentlyaccurate, independently of the traditional condition number of thematrix.

We characterize the set of nonsingular TN-quasiseparable matricesthrough the quasiseparable generators, the entries and thebidiagonal factorizations. This extends Gantmacher and Krein’s result.

We briefly mention other results on accurate computations with Green’squasiseparable matrices.

Goals of the talk

Relation with recent work by Gemignani

L. Gemignani, Neville elimination for rank-structured matrices, LAA2008.

We develop error analysis and Gemignani does not.

Gemignani deals with general (nL, nU )-order quasiseparable matrix andwe only with (1, 1)-order.

Due to this, we are able to present very simple and explicit conditions forTotal Nonnegativity.

We present a very simple and explicit algorithm in terms of parametersthat it is the key for error analysis.

Outline

2 Goals of the talk

Brief summary on Neville elimination (I)

It is a classic procedure to create zeros in a matrix by adding to arow (resp. column) a multiple of the previous row (resp. column).

It is valid for any matrix, it is different than Gaussian elimininationand it has different backward errors.

Without interchanges, it was carefully analyzed by Gasca and Peña ina series of seminal papers in the 90s. In particular, its matricialdescription in terms of bidiagonal factorizations and its fundamentalrelationship with total nonnegativity were established.

Theorem (Gasca and Peña (1994))

A nonsingular matrix A is TN if and only if complete Neville elimination can beperformed on A without row or column exchanges, with nonnegativemultipliers and positive diagonal pivots.

Brief summary on Neville elimination (II)

Neville elimination without exchanges adapts very well to thequasiseparable structure (Gemignani 2008).

The quasiseparable structure is destroyed by row or column exchanges.

In this talk, Neville elimination is never applied numerically. It is atheoretical way to get formulae, in terms of the generators, for thebidiagonal factors of the matrix. These formulae are then used

1 to develop fast algorithms,2 to perform their error analysis,3 and, to establish simple characterization of TN-quasiseparable

matrices.

Bidiagonal factorizations (I)

If complete Neville elimination runs without exchanges on a nonsingularn× n square matrix A then

A = L(1)L(2) · · ·L(n−1)DU (n−1) · · ·U (2)U (1),

where D is diagonal and

L(k) =

1. . .`(k)1 1

`(k)2 1

. . . . . .`(k)k 1

This is known as a bidiagonal factorization of A and all the factorsamount to n2 nontrivial entries.

Bidiagonal factorizations (I)

If complete Neville elimination runs without exchanges on a nonsingularn× n square matrix A then

A = L(1)L(2) · · ·L(n−1)DU (n−1) · · ·U (2)U (1),

where D is diagonal and

U (k) =

1. . . u

1 u(k)2

1. . .. . . u

This is known as a bidiagonal factorization of A and all the factorsamount to n2 nontrivial entries.

Bidiagonal factorizations (II)

Example (Bidiagonal factorization of a 5× 5 matrix)

A = L(1)L(2)L(3)L(4)DU (4)U (3)U (2)U (1),

×××××

A = L(1)L(2)L(3)L(4)DU(4)U (3)U (2)U (1),

L(4) =

1× 1× 1× 1× 1

U(4) =

1 ×1 ×

A = L(1)L(2)L(3)L(4)DU (4)U(3)U (2)U (1),

L(3) =

1× 1× 1× 1

U(3) =

1 ×1 ×

A = L(1)L(2)L(3)L(4)DU (4)U (3)U(2)U (1),

L(2) =

11× 1× 1

U(2) =

A = L(1)L(2)L(3)L(4)DU (4)U (3)U (2)U(1),

L(1) =

U(1) =

Simplifying assumptions

The generators of a quasiseparable matrix are not unique.

We will assume (not essential) that for any C quasiseparable matrix

C(i, 1 : i− 1) = 0 =⇒ pi = 0 and C(1 : i− 1, i) = 0 =⇒ hi = 0

Example (5× 5 quasiseparable matrix )

C(4, 1 : 3) =

[p4a3a2q1 p4a3q2 p4q3

]= 0 =⇒ p4 = 0

Similar for Green’s quasiseparable matrices.

C(i, 1 : i− 1) = 0 =⇒ pi = 0 and C(1 : i− 1, i) = 0 =⇒ hi = 0

C(4, 1 : 3) =

]= 0 =⇒ p4 = 0

C(i, 1 : i− 1) = 0 =⇒ pi = 0 and C(1 : i− 1, i) = 0 =⇒ hi = 0

C(4, 1 : 3) =

]= 0 =⇒ p4 = 0

C(i, 1 : i− 1) = 0 =⇒ pi = 0 and C(1 : i− 1, i) = 0 =⇒ hi = 0

C(4, 1 : 3) =

]= 0 =⇒ p4 = 0

C(i, 1 : i− 1) = 0 =⇒ pi = 0 and C(1 : i− 1, i) = 0 =⇒ hi = 0

C(1 : 3, 4) =

[g1b2b3h4 g2b3h4 g3h4

]T = 0 =⇒ h4 = 0

One more piece of standard notation...

Ei(α) =

1. . .

. . .1

, where α is in (i, i− 1) entry

RemarkIn the rest of the talk, we assume that Neville elimination runswithout exchanges. This is not a restriction for nonsingular TNmatrices.

One more piece of standard notation...

Ei(α) =

1. . .

. . .1

, where α is in (i, i− 1) entry

RemarkIn the rest of the talk, we assume that Neville elimination runswithout exchanges. This is not a restriction for nonsingular TNmatrices.

Neville on Green’s quasiseparable: E5(−`5)G = G4

(=g51g41

)if p4 6= 0

0 if p4 = 0→ E5(−`5) =

1−`5 1

rank one matrix ⇒ g51

g41=g52g42

=g53g43

=g54g44

Neville on Green’s quasiseparable: E5(−`5)G = G4

(=g51g41

)if p4 6= 0

0 if p4 = 0→ E5(−`5) =

1−`5 1

E5(−`5)G =

0 0 0 0 g′5h5

p5q5 = g5h5 =⇒ g′5 = g5 − `5g4b4

Bidiagonal factorization of Green’s quasiseparable

TheoremComplete Neville elimination runs without exchanges on a nonsingular n× nGreen’s quasiseparable matrix G specified by its generators if and only if

G = En(`n) · · ·E3(`3)E2(`2)DE2(u2)T E3(u3)T · · ·En(un)T ,

where D = diag(d1, . . . ,dn), and

`i :={ piai−1

pi−1if pi−1 6= 0

0 if pi−1 = 0ui :=

{hibi−1hi−1

if hi−1 6= 00 if hi−1 = 0

d1 = p1q1, di = piqi − `i ui pi−1qi−1 for i = 2 : n

Remarks

The bidiagonal factorization of G is sparse: 3n− 2 nontrivial entries

The bidiagonal factorization of G can be computed through explicitformulae from generators (also from entries) in O(n) flops.

Bidiagonal factorization of Green’s quasiseparable

TheoremComplete Neville elimination runs without exchanges on a nonsingular n× nGreen’s quasiseparable matrix G specified by its generators if and only if

G = En(`n) · · ·E3(`3)E2(`2)DE2(u2)T E3(u3)T · · ·En(un)T ,

where D = diag(d1, . . . ,dn), and

`i :={ piai−1

pi−1if pi−1 6= 0

0 if pi−1 = 0ui :=

{hibi−1hi−1

if hi−1 6= 00 if hi−1 = 0

d1 = p1q1, di = piqi − `i ui pi−1qi−1 for i = 2 : n

Remarks

The bidiagonal factorization of G is sparse: 3n− 2 nontrivial entries

The bidiagonal factorization of G can be computed through explicitformulae from generators (also from entries) in O(n) flops.

Neville on quasiseparable: E5(−`5)C = C4

(=c51c41

)if p4 6= 0

0 if p4 = 0→ E5(−`5) =

1−`5 1

× × × × ×× × × × ×× × × × ×× × × × ×× × × × ×

rank one matrix ⇒ c51

c41=c52c42

=c53c43

Neville on quasiseparable: E5(−`5)C = C4

(=c51c41

)if p4 6= 0

0 if p4 = 0→ E5(−`5) =

1−`5 1

E5(−`5)C =

× × × × ×× × × × ×× × × × ×× × × × ×0 0 0 × ×

Bidiagonal factorization of a quasiseparable matrix (I)

TheoremComplete Neville elimination runs without exchanges on a nonsingular n× nquasiseparable matrix C specified by its generators if and only if

C = En(`n) · · ·E4(`4)E3(`3)T E3(u3)T E4(u4)T · · ·En(un)T ,

`i :={ piai−1

pi−1if pi−1 6= 0

0 if pi−1 = 0ui :=

{hibi−1hi−1

if hi−1 6= 00 if hi−1 = 0

y1 z2x2 y2 z3

. . . . . . . . .xn−1 yn−1 zn

,has LDU factorization, whereF. M. Dopico (U. Carlos III, Madrid) Totally Nonnegative Quasiseparable Matrices POSTA 09 26 / 45

Bidiagonal factorization of a quasiseparable matrix (II)

Theorem (continued)

x2 = p2q1, xj = pj qj−1 − `j dj−1 for j = 3 : ny1 = d1, y2 = d2, yj = dj − `j gj−1 hj − uj pj qj−1 + uj `j dj−1 for j = 3 : n

z2 = g1h2, zj = gj−1 hj − ujdj−1 for j = 3 : n

Remarks

We can compute through formulae from the generators (or entries)

but, to get the bidiagonal factorization, it remains to compute withusual Gaussian (Neville) elimination on a tridiagonal matrix,

T = L(n−1)DU (n−1).

Total cost is O(n) flops and the bidiagonal factorization of C issparse: 5n− 6 nontrivial entries

Bidiagonal factorization of a quasiseparable matrix (II)

Theorem (continued)

Remarks

We can compute through formulae from the generators (or entries)

but, to get the bidiagonal factorization, it remains to compute withusual Gaussian (Neville) elimination on a tridiagonal matrix,

T = L(n−1)DU (n−1).

Total cost is O(n) flops and the bidiagonal factorization of C issparse: 5n− 6 nontrivial entries

Outline

2 Goals of the talk

TN quasiseparable and bidiagonal factorizations

TheoremAn n× n nonsingular matrix C is TN and quasiseparable if and only if

C = En(`n) · · ·E3(`3)L(n−1)DU (n−1)E3(u3)T · · ·En(un)T ,

with all the bidiagonal factors nonnegative and the diagonal entries of Dpositive.

TheoremAn n× n nonsingular matrix G is TN and Green’s quasiseparable if andonly if

G = En(`n) · · ·E2(`2)DE2(u2)T · · ·En(un)T ,

TN quasiseparable and bidiagonal factorizations

TheoremAn n× n nonsingular matrix C is TN and quasiseparable if and only if

TheoremAn n× n nonsingular matrix G is TN and Green’s quasiseparable if andonly if

G = En(`n) · · ·E2(`2)DE2(u2)T · · ·En(un)T ,

TN-Green’s quasiseparable matrices (I)

Theorem (characterization in terms of the parameters)

Let G be an n× n Green’s quasiseparable matrix specified by its generators.Then

G is nonsingular and TN⇐⇒

p1 q1 > 0, and

pi qi −(piai−1

pi−1

) (hibi−1

hi−1

)pi−1 qi−1 > 0,

piai−1

pi−1≥ 0,

hibi−1

hi−1≥ 0, for 2 ≤ i ≤ n

These conditions can be checked in O(n) flops.

TN-Green’s quasiseparable matrices (II)

Theorem (characterization in terms of the entries)

Let G be an n× n Green’s quasiseparable matrix. Then

gii > 0, 1 ≤ i ≤ ngi,i−1 ≥ 0, gi−1,i ≥ 0, 2 ≤ i ≤ ndetG(i− 1 : i , i− 1 : i) > 0, 2 ≤ i ≤ n

× × × × ×× × × × ×× × × × ×× × × × ×× × × × ×

TN-quasiseparable matrices (I)

TheoremLet C be an n× n quasiseparable matrix specified by its generators. Define

`i :={ piai−1

pi−1if pi−1 6= 0

0 if pi−1 = 0ui :=

{hibi−1hi−1

if hi−1 6= 00 if hi−1 = 0

for i = 3 : n,

y1 z2x2 y2 z3

. . . . . . . . .xn−1 yn−1 zn

TN-quasiseparable matrices (II)

Theorem (continued)

Then, C is nonsingular and TN if and only if

`i ≥ 0 and ui ≥ 0 for i = 3 : n.

The tridiagonal matrix T is nonsingular and TN.

If pj qj−1 = 0, for some j, then C(j : n , 1 : j − 1) = 0.

If gj−1 hj = 0, for some j, then C(1 : j − 1 , j : n) = 0.

p4a3a2q1 p4a3q2 p4q3 = 0 d4 g4h5

Theorem (continued)

`i ≥ 0 and ui ≥ 0 for i = 3 : n.

The tridiagonal matrix T is nonsingular and TN.

p4a3a2q1 p4a3q2 p4q3 = 0 d4 g4h5

Theorem (continued)

`i ≥ 0 and ui ≥ 0 for i = 3 : n.

The tridiagonal matrix T in nonsingular and TN.

p4a3a2q1 p4a3q2 p4q3 = 0 d4 g4h5

p5a4a3a2q1 p5a4a3q2 0 p5q4 d5

Theorem (continued)

`i ≥ 0 and ui ≥ 0 for i = 3 : n.

0 0 p4q3 = 0 d4 g4h5

Theorem (continued)

`i ≥ 0 and ui ≥ 0 for i = 3 : n.

0 0 p4q3 = 0 d4 g4h5

0 0 0 p5q4 d5

Outline

2 Goals of the talk

Given a bidiagonal factorization...Assume that for a general matrix A:

We know a bidiagonal factorization

A = L(1)L(2) · · ·L(n−1)DU (n−1) · · ·U (2)U (1).

We want to solve Ax = b.

L(1)x(1) = b, L(2)x(2) = x(1), · · · L(n−1)x(n−1) = x(n−2),

Dx(n) = x(n−1),

U (n−1)x(n+1) = x(n), U (n−2)x(n+2) = x(n+1), · · · U (1)x = x(2n−2)

Observe that for quasiseparable matrices many of the bidiagonalsystems are very simple and can be solved in two flops:

Ei(α)z = y ⇐⇒ z = Ei(−α)y.

In fact, all are of this type for Green’s quasiseparable matrices.F. M. Dopico (U. Carlos III, Madrid) Totally Nonnegative Quasiseparable Matrices POSTA 09 35 / 45

A = L(1)L(2) · · ·L(n−1)DU (n−1) · · ·U (2)U (1).

L(1)x(1) = b, L(2)x(2) = x(1), · · · L(n−1)x(n−1) = x(n−2),

Dx(n) = x(n−1),

Ei(α)z = y ⇐⇒ z = Ei(−α)y.

A = L(1)L(2) · · ·L(n−1)DU (n−1) · · ·U (2)U (1).

L(1)x(1) = b, L(2)x(2) = x(1), · · · L(n−1)x(n−1) = x(n−2),

Dx(n) = x(n−1),

Ei(α)z = y ⇐⇒ z = Ei(−α)y.

A = L(1)L(2) · · ·L(n−1)DU (n−1) · · ·U (2)U (1).

L(1)x(1) = b, L(2)x(2) = x(1), · · · L(n−1)x(n−1) = x(n−2),

Dx(n) = x(n−1),

Ei(α)z = y ⇐⇒ z = Ei(−α)y.

A = L(1)L(2) · · ·L(n−1)DU (n−1) · · ·U (2)U (1).

L(1)x(1) = b, L(2)x(2) = x(1), · · · L(n−1)x(n−1) = x(n−2),

Dx(n) = x(n−1),

Ei(α)z = y ⇐⇒ z = Ei(−α)y.

The complete O(n) quasiseparable algorithm

ALGORITHM 1

INPUT: Generators of C (resp. G) n× n quasiseparable (resp. Green’squasiseparable) matrix and vector b

OUTPUT: Solution of Cx = b (resp. Gx = b)

Compute bidiagonal factorization with formulae as in the first part ofthe talk:

(resp. G = En(`n) · · ·E2(`2)DE2(u2)T · · ·En(un)T )

Solve a sequence of bidiagonal systems to get x as in the previousslide.

Outline

2 Goals of the talk

Backward errors for general quasiseparable matrix

TheoremIf Algorithm 1 is applied to solve Cx = b, where C is n× n quasiseparablematrix, and

En(n), . . . , E3(3), L(n−1), D, U (n−1), E3(u3)T , . . . , En(un)T ,

are the computed bidiagonal factors of C with unit roundoff ε, then thecomputed solution x satisfies

(C + E)x = b,

(C + E) is quasiseparable, and

|E| ≤ 27nε1− 27nε

En(|n|) · · ·E3(|3|) |L(n−1)| |D| |U (n−1)|E3(|u3|)T · · ·En(|un|)T

Comments on this backward error analysis

Similar result for Green’s quasiseparable matrices, preserving theGreen’s structure.

It is very tricky.

It requires a delicate way to evaluate the formulae for thebidiagonal/tridiagonal factors

It combines1 mixed backward-forward errors in terms of parameters and

bidiagonal factors, with2 backward errors in terms of entries.

The bound may be not satisfactory if

En(|n|) · · ·E3(|3|) |L(n−1)| |D| |U (n−1)|E3(|u3|)T · · ·En(|un|)T >> |C|

No way to incorporate pivoting to improve bounds because it destroysquasiseparable structure and it does not match well Neville elimination.

It is very tricky.

Satisfactory backward errors for TN-quasiseparable matrices

TheoremIf Algorithm 1 is applied to solve Cx = b, where C is n× n quasiseparablematrix, and all the computed bidiagonal factors of C are nonnegative(diag D > 0), then the computed solution x satisfies

(C + E)x = b,

(C + E) is TN-quasiseparable, and

|E| ≤ 27nε1− 54nε

Satisfactory backward errors for TN-quasiseparable matrices

TheoremIf Algorithm 1 is applied to solve Cx = b, where C is n× n quasiseparablematrix, and all the computed bidiagonal factors of C are nonnegative(diag D > 0), then the computed solution x satisfies

(C + E)x = b,

(C + E) is TN-quasiseparable, and

|E| ≤ 27nε1− 54nε

Forward errors for TN-Green’s quasiseparable (I)

TheoremLet G be an n× n Green’s quasiseparable matrix and define

κGQ(G) = max2≤i≤n

|gi,i gi−1,i−1|+ |gi,i−1 gi−1,i||gi,i gi−1,i−1 − gi,i−1 gi−1,i|

Assume that Algorithm 1 is applied to solve Gx = b with unit roundoff ε andthat the computed bidiagonal factors of G are nonnegative (Dnonsingular). If

9ε1− 9ε

κGQ(G) <12,

G is nonsingular and TN.

The computed solution x satisfies

|x− x| ≤ 2(

8nε1− 8nε

+ κGQ(G)9ε

1− 9ε

)|G−1||b|

Forward errors for TN-Green’s quasiseparable (II)

|x− x| ≤ 2(

8nε1− 8nε

+ κGQ(G)9ε

1− 9ε

)|G−1||b|

If ε κGQ(G) << 1, this is a very satisfactory bound, because

‖ |G−1||b| ‖∞‖x‖∞

=‖ |G−1||b| ‖∞‖G−1b‖∞

is moderate except for particular b’s.

is a condition number for this problem.

Note that

detG(i− 1 : i , i− 1 : i) = gi,i gi−1,i−1 − gi,i−1 gi−1,i.

|x− x| ≤ 2(

8nε1− 8nε

+ κGQ(G)9ε

1− 9ε

)|G−1||b|

‖ |G−1||b| ‖∞‖x‖∞

=‖ |G−1||b| ‖∞‖G−1b‖∞

Note that

|x− x| ≤ 2(

8nε1− 8nε

+ κGQ(G)9ε

1− 9ε

)|G−1||b|

‖ |G−1||b| ‖∞‖x‖∞

=‖ |G−1||b| ‖∞‖G−1b‖∞

Note that

|x− x| ≤ 2(

8nε1− 8nε

+ κGQ(G)9ε

1− 9ε

)|G−1||b|

‖ |G−1||b| ‖∞‖x‖∞

=‖ |G−1||b| ‖∞‖G−1b‖∞

Note that

|x− x| ≤ 2(

8nε1− 8nε

+ κGQ(G)9ε

1− 9ε

)|G−1||b|

‖ |G−1||b| ‖∞‖x‖∞

=‖ |G−1||b| ‖∞‖G−1b‖∞

Note that

Other results for Green’s quasiseparable

We know how to compute in O(n2) flops eigenvalues andsingular values of TN-Green’s quasiseparable matrices withrelative errors

O(ε κGQ(G))

We have shown perfect componentwise backward stability insolving linear systems trough bidiagonalization for diagonallydominant Green’s quasiseparable matrices.

Other results for Green’s quasiseparable

We know how to compute in O(n2) flops eigenvalues andsingular values of TN-Green’s quasiseparable matrices withrelative errors

O(ε κGQ(G))

We have shown perfect componentwise backward stability insolving linear systems trough bidiagonalization for diagonallydominant Green’s quasiseparable matrices.

Outline

2 Goals of the talk

Conclusions and future work

Bidiagonalization + solving a sequence of bidiagonal systems isfast on quasiseparable but not backward stable.It is fast and backward stable on TN-quasiseparable.Simple forward error bounds for TN Green’s quasiseparableavailable.Next step: error analysis of quasiseparable structured QRfactorization for solving linear systems.Next step: Fast and accurate computations of eigenvalues andsingular values of quasiseparable matrices given the sparsebidiagonal factorization.Error analysis of structured rank algorithms is an open areafull of challenging problems....

fast and accurate computations with totally nonnegative ... · fast and accurate computations with...

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