Blocked Algorithms for the Matrix SquareRoot

Edvin Deadman 1 Nick Higham 2 Rui Ralha 3

1University of Manchester / Numerical Algorithms Group

2University of Manchester

3University of Minho, Portugal

May 18, 2012

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 1/29

Outline

1. Numerical Libraries and NAG2. The Matrix Square Root:

Definition and UsesThe Schur MethodBlockingParallelism

3. Implementation of Algorithms in Numerical Libraries

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 2/29

Why use numerical libraries?

Numerical computation is difficult to do accurately.Writing routines from scratch is time consuming.Commonly encountered problems:

Overflow/underflow: how does the computer deal with large/ small numbers?Condition: how sensitive is the solution to small changes inthe input?Stability: how sensitive is the computation to roundingerrors?

Important considerations:Error analysis.Information about error bounds.Parallelism.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 3/29

How a numerical library is used

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 4/29

The NAG Library

Root FindingSummation of SeriesQuadratureOrdinary Differential EquationsPartial Differential EquationsNumerical DifferentiationIntegral EquationsMesh GenerationInterpolationCurve and Surface FittingOptimizationApproximations of SpecialFunctions

Dense Linear AlgebraSparse Linear AlgebraCorrelation & Regression AnalysisMultivariate MethodsAnalysis of VarianceRandom Number GeneratorsUnivariate EstimationNonparametric StatisticsSmoothing in StatisticsContingency Table AnalysisSurvival AnalysisTime Series AnalysisOperations Research

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 5/29

Defining the Matrix Square Root

A square root of A is a solution of X 2 = A.

There can be infinitely many square roots. They are neverunique: (

cos θ sin θsin θ − cos θ

)2=

(1 00 1

).

If A has m Jordan blocks and s ≤ m distinct eigenvaluesthen there are 2s square roots that are primary functions ofA. [5]

If A has no eigenvalues on the negative real line, thenthere is a unique principal square root A1/2 witheigenvalues in the right half plane.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 6/29

Defining the Matrix Square Root

A square root of A is a solution of X 2 = A.There can be infinitely many square roots. They are neverunique: (

cos θ sin θsin θ − cos θ

)2=

(1 00 1

).

If A has m Jordan blocks and s ≤ m distinct eigenvaluesthen there are 2s square roots that are primary functions ofA. [5]

If A has no eigenvalues on the negative real line, thenthere is a unique principal square root A1/2 witheigenvalues in the right half plane.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 6/29

Defining the Matrix Square Root

A square root of A is a solution of X 2 = A.There can be infinitely many square roots. They are neverunique: (

cos θ sin θsin θ − cos θ

)2=

(1 00 1

).

If A has m Jordan blocks and s ≤ m distinct eigenvaluesthen there are 2s square roots that are primary functions ofA. [5]

If A has no eigenvalues on the negative real line, thenthere is a unique principal square root A1/2 witheigenvalues in the right half plane.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 6/29

Defining the Matrix Square Root

A square root of A is a solution of X 2 = A.There can be infinitely many square roots. They are neverunique: (

cos θ sin θsin θ − cos θ

)2=

(1 00 1

).

If A has m Jordan blocks and s ≤ m distinct eigenvaluesthen there are 2s square roots that are primary functions ofA. [5]

If A has no eigenvalues on the negative real line, thenthere is a unique principal square root A1/2 witheigenvalues in the right half plane.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 6/29

Defining the Matrix Square Root: Example

A =

1 1 0 00 1 0 00 0 2 10 0 0 2

Four primary square roots:1 12 0 00 1 0 00 0

√2 1

2√

20 0 0

√2

,−1 −12 0 00 −1 0 00 0 −

√2 − 1

2√

20 0 0 −

√2

,

1 12 0 00 1 0 00 0 −

√2 − 1

2√

20 0 0 −

√2

,−1 −12 0 00 −1 0 00 0

√2 1

2√

20 0 0

√2

.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 7/29

Defining the Matrix Square Root: Example

A =

1 1 0 00 1 0 00 0 2 10 0 0 2

Four primary square roots:

1 12 0 00 1 0 00 0

√2 1

2√

20 0 0

√2

,−1 −12 0 00 −1 0 00 0 −

√2 − 1

2√

20 0 0 −

√2

,

1 12 0 00 1 0 00 0 −

√2 − 1

2√

20 0 0 −

√2

,−1 −12 0 00 −1 0 00 0

√2 1

2√

20 0 0

√2

.Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, Portugal

Blocked Algorithms for the Matrix Square Root 7/29

Defining the Matrix Square Root: Example

A =

1 1 0 00 1 0 00 0 2 10 0 0 2

Four primary square roots:

1 12 0 00 1 0 00 0

√2 1

2√

20 0 0

√2

,−1 −12 0 00 −1 0 00 0 −

√2 − 1

2√

20 0 0 −

√2

,

1 12 0 00 1 0 00 0 −

√2 − 1

2√

20 0 0 −

√2

,−1 −12 0 00 −1 0 00 0

√2 1

2√

20 0 0

√2

.Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, Portugal

Blocked Algorithms for the Matrix Square Root 8/29

Uses of the Matrix Square Root

Markov models of finance and population dynamics:If P(t) is the transition matrix for a time step t , then

√P

could be used for t2 .Roots of stochastic matrices: still an open problem.

Solution of differential equations:ÿ + Ay = 0 has solution y = cos(

√At)y0 + sin(

√At)y1.

Polar decomposition / matrix sign decomposition.Important kernel routine for computing matrix logarithm,pth roots and powers and trigonometric matrixfunctions. [1;6]

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 9/29

Uses of the Matrix Square Root

Markov models of finance and population dynamics:If P(t) is the transition matrix for a time step t , then

√P

could be used for t2 .Roots of stochastic matrices: still an open problem.

Solution of differential equations:ÿ + Ay = 0 has solution y = cos(

√At)y0 + sin(

√At)y1.

Polar decomposition / matrix sign decomposition.Important kernel routine for computing matrix logarithm,pth roots and powers and trigonometric matrixfunctions. [1;6]

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 9/29

Uses of the Matrix Square Root

Markov models of finance and population dynamics:If P(t) is the transition matrix for a time step t , then

√P

could be used for t2 .Roots of stochastic matrices: still an open problem.

Solution of differential equations:ÿ + Ay = 0 has solution y = cos(

√At)y0 + sin(

√At)y1.

Polar decomposition / matrix sign decomposition.

Important kernel routine for computing matrix logarithm,pth roots and powers and trigonometric matrixfunctions. [1;6]

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 9/29

Uses of the Matrix Square Root

Markov models of finance and population dynamics:If P(t) is the transition matrix for a time step t , then

√P

could be used for t2 .Roots of stochastic matrices: still an open problem.

Solution of differential equations:ÿ + Ay = 0 has solution y = cos(

√At)y0 + sin(

√At)y1.

Polar decomposition / matrix sign decomposition.Important kernel routine for computing matrix logarithm,pth roots and powers and trigonometric matrixfunctions. [1;6]

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 9/29

The Schur Method

1. Compute a Schur decomposition: A = QTQ∗ with T uppertriangular. [2]

2. Expand U2 = T elementwise. For a primary square root, Uis also upper triangular:

U2ii = Tii ,

UiiUij + UijUjj = Tij −j−1∑

k=i+1

UikUkj .

U is found either a column or a superdiagonal at a time.3.√

A = QUQ∗

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 10/29

The Schur Method: Properties

Cost: 2813n3 flops.

Real arithmetic version uses quasi upper triangular Schurdecomposition and 2× 2 diagonal block structure. [4]

Computed square root of the triangular matrix Û satisfiesÛ2 = T + ∆T , where

|∆T | ≤ γ̃n|Û|2

(this only holds normwise in the real arithmetic version).

No use of level 3 BLAS - slow!Now focus on the triangular phase of the algorithm.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 11/29

The Schur Method: Properties

Cost: 2813n3 flops.

Real arithmetic version uses quasi upper triangular Schurdecomposition and 2× 2 diagonal block structure. [4]

Computed square root of the triangular matrix Û satisfiesÛ2 = T + ∆T , where

|∆T | ≤ γ̃n|Û|2

(this only holds normwise in the real arithmetic version).No use of level 3 BLAS - slow!Now focus on the triangular phase of the algorithm.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 11/29

Run times for random complex triangular matrices

0 1000 2000 3000 4000 5000 6000 7000 80000

50

100

150

200

250

300

350

400

450

n

time

(s)

point

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 12/29

The Blocked Schur Method

The Uij and Tij are now taken to be blocks:

U2ii = Tii , (1)

UiiUij + UijUjj = Tij −j−1∑

k=i+1

UikUkj . (2)

Solve (1) using the point method and (2) by solving theSylvester equation (e.g. xTRSYL in LAPACK).

Same error bounds hold true.Over 90% of run time spent in GEMM calls and 8% inSylvester equation solution.Not very sensitive to block size or choice of kernel routines.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 13/29

The Blocked Schur Method

The Uij and Tij are now taken to be blocks:

U2ii = Tii , (1)

UiiUij + UijUjj = Tij −j−1∑

k=i+1

UikUkj . (2)

Solve (1) using the point method and (2) by solving theSylvester equation (e.g. xTRSYL in LAPACK).Same error bounds hold true.Over 90% of run time spent in GEMM calls and 8% inSylvester equation solution.Not very sensitive to block size or choice of kernel routines.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 13/29

Run times for random complex triangular matrices

0 1000 2000 3000 4000 5000 6000 7000 80000

50

100

150

200

250

300

350

400

450

n

time

(s)

pointblock

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 14/29

The Recursive Blocked Schur Method

Recursive solution of the triangular phase: [3](U11 U120 U22

)2=

(T11 T120 T22

).

U211 = T11 and U222 = T22 are solved recursively.

Solve U11U12 + U12U22 = T12 using the recursive methodof Jonsson & Kågström. [7]

‘Point’ algorithms are used when the recursion hasreached a certain size (e.g. n = 64).Same error bound holds as point algorithm (proof byinduction).

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 15/29

Run times for random complex triangular matrices

0 1000 2000 3000 4000 5000 6000 7000 80000

50

100

150

200

250

300

350

400

450

n

time

(s)

pointblockrecursion

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 16/29

Aside: multiplying two triangular matrices

0 1000 2000 3000 4000 5000 6000 7000 80000

50

100

150

200

250

n

time

(s)

pointblockrecursion

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 17/29

Full complex matrices (called from MATLAB)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

n

time

(s)

sqrtmfort_pointfort_recurse

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 18/29

Full real matrices (called from MATLAB)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

n

time

(s)

sqrtm_realsqrtmfort_point_realfort_recurse_real

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 19/29

Parallelism

Which approach is best in parallel?

Options for parallelism:Threaded BLAS.Explicit loop-based parallelism of the triangular phase.OpenMP tasks.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 20/29

Parallelism

Which approach is best in parallel?

Options for parallelism:Threaded BLAS.Explicit loop-based parallelism of the triangular phase.OpenMP tasks.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 20/29

Parallel Point and Block Methods

Blocks below and left of (i , j) block must be computed first:

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 21/29

Parallel Point and Block Methods

Synchronisation required after each superdiagonal:

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 22/29

Parallel Point and Block Methods

Synchronisation required after each superdiagonal:

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 22/29

Parallel Point and Block Methods

Synchronisation required after each superdiagonal:

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 22/29

Parallel Point and Block Methods

Synchronisation required after each superdiagonal:

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 22/29

Recursive Blocking in Parallel

Task based approach - each recursive call generates newtasks. Synchronization points required:

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 23/29

Recursive Blocking in Parallel

Task based approach - each recursive call generates newtasks. Synchronization points required:

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 23/29

Recursive Blocking in Parallel

Task based approach - each recursive call generates newtasks. Synchronization points required:

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 23/29

Recursive Blocking in Parallel

Task based approach - each recursive call generates newtasks. Synchronization points required:

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 23/29

Recursive Blocking in Parallel

Task based approach - each recursive call generates newtasks. Synchronization points required:

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 23/29

Parallel Results for 4000× 4000 Triangular Matrices

1 Thread 8 Threads − BLAS 8 Threads − OpenMP0

20

40

60

80

100

120

140tim

e (s

)

pointblockrecursion

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 24/29

Parallel Results for 4000× 4000 Full Matrices

1 Thread 8 Threads − BLAS 8 Threads − OpenMP0

100

200

300

400

500

600

700

800

900

1000tim

e (s

)

pointblockrecursion

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 25/29

Implementation for the NAG Library

Key: robustness and error handling.

Extension to singular matrices:Eigenvalue considered to vanish if λ < �‖A‖Reorder and check if the vanishing eigenvalue issemisimple.

Negative eigenvalues:Eigenvalue considered to lie on R− if Re(λ) < 0 and|Im(λ)| < �|Re(λ)|.A non-principal square root can be returned in this case.

Condition estimation:Condition number estimates and residual bounds areavailable for the matrix square root.Condition number is expensive to compute.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 26/29

Implementation for the NAG Library

Key: robustness and error handling.Extension to singular matrices:

Eigenvalue considered to vanish if λ < �‖A‖Reorder and check if the vanishing eigenvalue issemisimple.

Negative eigenvalues:Eigenvalue considered to lie on R− if Re(λ) < 0 and|Im(λ)| < �|Re(λ)|.A non-principal square root can be returned in this case.

Condition estimation:Condition number estimates and residual bounds areavailable for the matrix square root.Condition number is expensive to compute.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 26/29

Implementation for the NAG Library

Key: robustness and error handling.Extension to singular matrices:

Eigenvalue considered to vanish if λ < �‖A‖Reorder and check if the vanishing eigenvalue issemisimple.

Negative eigenvalues:Eigenvalue considered to lie on R− if Re(λ) < 0 and|Im(λ)| < �|Re(λ)|.A non-principal square root can be returned in this case.

Condition estimation:Condition number estimates and residual bounds areavailable for the matrix square root.Condition number is expensive to compute.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 26/29

Implementation for the NAG Library

Key: robustness and error handling.Extension to singular matrices:

Eigenvalue considered to vanish if λ < �‖A‖Reorder and check if the vanishing eigenvalue issemisimple.

Negative eigenvalues:Eigenvalue considered to lie on R− if Re(λ) < 0 and|Im(λ)| < �|Re(λ)|.A non-principal square root can be returned in this case.

Condition estimation:Condition number estimates and residual bounds areavailable for the matrix square root.Condition number is expensive to compute.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 26/29

Implementation for the NAG Library

Real and complex routines, with or without conditionestimation, and extra routines specifically for triangularmatrices.Test programs:

Check the residual (√

A)2 − A against the theoreticalresidual bound.Test against computation in VPA using condition estimate.Tricky test cases e.g. almost singular matrices, nearlynegative eigenvalues.Error exits and illegal inputs.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 27/29

Summary

Recursive blocking can be fast, when the algorithm allows.NAG implementation uses OpenMP, so will use standardblocking.Matrix square root is a key computational kernel - BLAS?The fastest method in serial is not necessarily the fastestmethod in parallel.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 28/29

References[1] A. H. Al-Mohy and N. J. Higham. Improved inverse scaling and

squaring algorithms for the matrix logarithm. MIMS Eprint, 83,2011.

[2] Å. Björck and S. Hammarling. A Schur method for the square rootof a matrix. Lin. Alg. & Appl., 52/53:127–140, 1983.

[3] E. Deadman, N. J. Higham, and R. Ralha. A recursive blockedSchur algorithm for computing the matrix square root. MIMSEprint, 26, 2012.

[4] N. J. Higham. Computing real square roots of a real matrix. Lin.Alg. & Appl., 88/89:405–430, 1987.

[5] N. J. Higham. Functions of Matrices: Theory and Computation.SIAM, 2008.

[6] N. J. Higham and L. Lin. A Schur-Padé algorithm for fractionalpowers of a matrix. SIAM J. Matrix Anal. & Appl.,32(3):1056–1078, 2010.

[7] I. Jonsson and B. Kågström. Recursive blocked algorithms forsolving triangular systems - part I: One-sided and coupledSylvester-type matrix equations. ACM Transactions onMathematical Software, 28(4):392–415, December 2002.

Edvin Deadman , Nick Higham , Rui Ralha , University of Manchester / Numerical Algorithms Group, University of Manchester, University of Minho, PortugalBlocked Algorithms for the Matrix Square Root 29/29