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LIST OF ALGORITHMS
LIST OF ALGORITHMS 3.3.1 3.4.1 3.4.2 3.6.1 3.8.1 3.9.1
Back Substitution Method for Upper Triangular System The Cholesky Algorithm LU Factorization of an Upper Hessenberg Matrix Givens QR Factorization Least Squares Solution Using QR Factorization Least Squares Solutions Using the SVD
38 51 52 62 65 72
4.5.1
Complete QZ Algorithm for Reduction to Generalized Schur Form
99
5.3.1 5.3.2 5.3.3 5.5.1
Pad6 Approximation to e A using Scaling and Squaring Schur Algorithm for Matrix Exponential Computing Integrals involving Matrix Exponential Hessenberg Algorithm for the Frequency Response Matrix
132 135 137 144
6.7.1 6.9.1 6.9.2
Staircase Algorithm Newton's Algorithm for Distance to Uncontrollability An Algorithm for Computing #(A, B)
174 186 189
7.2.1 7.5.1 7.6.1
Computing the Hz-Norm Computing Inertia and Stability Bisection Algorithm for Estimating Distance to Continuous-time Instability Bisection Algorithm for Estimating Distance to a Discrete-Unstable System
212 219
8.3.1 8.5.1 8.6.1 8.6.2
sep Estimation The Hessenberg-Schur Algorithm for XA + B X = C Cholesky Factor for Continuous Lyapunov Equation Cholesky Factor for Discrete Lyapunov Equation
261 270 286 290
9.3.1 9.3.2 9.4.1
An SVD Algorithm for Minimal Realization A Modified SVD Algorithm for Minimal Realization A Deterministic Subspace Identification Algorithm
317 322 327
7.6.2
225 229
oo~ XXXUl
xxxiv
LIST OF ALGORITHMS
9.4.2 9.4.3
A Subspace Stochastic Identification Algorithm Continuous-Time Frequency-Domain Subspace Identification
330 333
10.5.1 10.6.1 10.6.2 10.7.1
The Algorithm Continuous-Time LQR Design Bisection Algorithm for Hoe-Norm Two-Step Algorithm for H~-Norm Bisection Method for Complex Stability Radius
366 376 379 390
11.2.1
The Recursive Algorithm for Singe-Input Hessenberg EVA Problem An RQ Implementation of the Recursive Algorithm A Storage-Efficient Version of the RQ Implementation The Recursive Algorithm for Multi-Input EVA Problem An Algorithm to Assign p(p = 1 or 2) Eigen values The Schur Algorithm for Multi-Input EVA Problem A Parametric Sylvester Equation Algorithm for PEVA Robust Eigenvalue Assignment Algorithm (The KNV Algorithm)
411 417 418 423 430 431 437 447
11.2.2 11.2.3
11.3.1 11.3.2
11.3.3 11.3.4 11.6.1
472 476
12.8.1 12.9.1 12.10.1
Full-Order Observer Design via Sylvester-Observer Equation Reduced-Order Observer Design via EVA Reduced-Order Observer Design via Sylvester-Observer Equation A Recursive Algorithm for the Multi-Output Sylvester-Observer Equation A Recursive Block Triangular Algorithm for the Multi-Output Sylvester-Observer Equation An Algorithm for Constrained Sylvester-Observer Equation State Estimation using Kalman Filter LQG Design for Continuous time System
13.5.1 13.5.2 13.5.3 13.5.4 13.5.5 13.5.6 13.5.7 13.5.8 13.5.9 13.5.10 13.5.11
Schur Algorithm for the CARE Generalized Schur Algorithm for the DARE Inverse-Free Generalized Schur Method for the CARE Inverse-Free Generalized Schur Algorithm for the DARE Computing Sign (A) Matrix Sign-Function Algorithm for the CARE Matrix Sign-Function Algorithm for the DARE Newton's Method for the CARE Newton's Method with Line Search for the CARE Newton's Method for the DARE Newton's Method with Line Search for the DARE
542 553 556 558 560 563 566 568 572 574 577
14.2.1 14.2.2
Internal Balancing of a Continuous-Time Minimal Realization Square-Root Algorithm for Internal Balancing of a Continuous-Time Nonminimal Realization
604
12.3.1 12.4.1 12.4.2 12.7.1 12.7.2
479 488 492 497 501 507
607
LIST OF A L G O R I T H M S
xxxv
14.4.1 14.4.2 14.5.1
Model Reduction via Balanced Truncation Schur Algorithm for Continuous-Time Model Reduction Hankel-Norm Approximation of Continuous-Time System
614 616 626
15.2.1 15.2.2 15.4.1 15.4.2
Block Arnoldi Algorithm Nonsymmetric Lanczos Algorithm Amoldi Algorithm for Single-input Lyapunov Equation Block Arnoldi Algorithm for Stable Discrete-Time Lyapunov Equation Restarted Amoldi Algorithm for Sylvester Equation Block Arnoldi Algorithm for Sylvester Equation Amoldi Algorithm for Single-Output Sylvester-Observer Equation Arnoldi Algorithm for CARE Projection Algorithm for Partial Pole-Placement Lanczos Algorithm for SISO Model Reduction Amoldi Algorithm for SISO Model Reduction
651 652 653
15.4.3 15.4.4 15.4.5 15.4.6 15.5.1 15.6.1 15.6.2
654 655 656 657 658 659 660 661
Back Substitution Method for Upper Triangular System The Cholesky Algorithm LU Factorization of an Upper Hessenberg Matrix Givens QR Factorization Least Squares Solution Using QR Factorization Least Squares Solutions Using the SVD
38 51 52 62 65 72
4.5.1
Complete QZ Algorithm for Reduction to Generalized Schur Form
99
5.3.1 5.3.2 5.3.3 5.5.1
Pad6 Approximation to e A using Scaling and Squaring Schur Algorithm for Matrix Exponential Computing Integrals involving Matrix Exponential Hessenberg Algorithm for the Frequency Response Matrix
132 135 137 144
6.7.1 6.9.1 6.9.2
Staircase Algorithm Newton's Algorithm for Distance to Uncontrollability An Algorithm for Computing #(A, B)
174 186 189
7.2.1 7.5.1 7.6.1
Computing the Hz-Norm Computing Inertia and Stability Bisection Algorithm for Estimating Distance to Continuous-time Instability Bisection Algorithm for Estimating Distance to a Discrete-Unstable System
212 219
8.3.1 8.5.1 8.6.1 8.6.2
sep Estimation The Hessenberg-Schur Algorithm for XA + B X = C Cholesky Factor for Continuous Lyapunov Equation Cholesky Factor for Discrete Lyapunov Equation
261 270 286 290
9.3.1 9.3.2 9.4.1
An SVD Algorithm for Minimal Realization A Modified SVD Algorithm for Minimal Realization A Deterministic Subspace Identification Algorithm
317 322 327
7.6.2
225 229
oo~ XXXUl
xxxiv
LIST OF ALGORITHMS
9.4.2 9.4.3
A Subspace Stochastic Identification Algorithm Continuous-Time Frequency-Domain Subspace Identification
330 333
10.5.1 10.6.1 10.6.2 10.7.1
The Algorithm Continuous-Time LQR Design Bisection Algorithm for Hoe-Norm Two-Step Algorithm for H~-Norm Bisection Method for Complex Stability Radius
366 376 379 390
11.2.1
The Recursive Algorithm for Singe-Input Hessenberg EVA Problem An RQ Implementation of the Recursive Algorithm A Storage-Efficient Version of the RQ Implementation The Recursive Algorithm for Multi-Input EVA Problem An Algorithm to Assign p(p = 1 or 2) Eigen values The Schur Algorithm for Multi-Input EVA Problem A Parametric Sylvester Equation Algorithm for PEVA Robust Eigenvalue Assignment Algorithm (The KNV Algorithm)
411 417 418 423 430 431 437 447
11.2.2 11.2.3
11.3.1 11.3.2
11.3.3 11.3.4 11.6.1
472 476
12.8.1 12.9.1 12.10.1
Full-Order Observer Design via Sylvester-Observer Equation Reduced-Order Observer Design via EVA Reduced-Order Observer Design via Sylvester-Observer Equation A Recursive Algorithm for the Multi-Output Sylvester-Observer Equation A Recursive Block Triangular Algorithm for the Multi-Output Sylvester-Observer Equation An Algorithm for Constrained Sylvester-Observer Equation State Estimation using Kalman Filter LQG Design for Continuous time System
13.5.1 13.5.2 13.5.3 13.5.4 13.5.5 13.5.6 13.5.7 13.5.8 13.5.9 13.5.10 13.5.11
Schur Algorithm for the CARE Generalized Schur Algorithm for the DARE Inverse-Free Generalized Schur Method for the CARE Inverse-Free Generalized Schur Algorithm for the DARE Computing Sign (A) Matrix Sign-Function Algorithm for the CARE Matrix Sign-Function Algorithm for the DARE Newton's Method for the CARE Newton's Method with Line Search for the CARE Newton's Method for the DARE Newton's Method with Line Search for the DARE
542 553 556 558 560 563 566 568 572 574 577
14.2.1 14.2.2
Internal Balancing of a Continuous-Time Minimal Realization Square-Root Algorithm for Internal Balancing of a Continuous-Time Nonminimal Realization
604
12.3.1 12.4.1 12.4.2 12.7.1 12.7.2
479 488 492 497 501 507
607
LIST OF A L G O R I T H M S
xxxv
14.4.1 14.4.2 14.5.1
Model Reduction via Balanced Truncation Schur Algorithm for Continuous-Time Model Reduction Hankel-Norm Approximation of Continuous-Time System
614 616 626
15.2.1 15.2.2 15.4.1 15.4.2
Block Arnoldi Algorithm Nonsymmetric Lanczos Algorithm Amoldi Algorithm for Single-input Lyapunov Equation Block Arnoldi Algorithm for Stable Discrete-Time Lyapunov Equation Restarted Amoldi Algorithm for Sylvester Equation Block Arnoldi Algorithm for Sylvester Equation Amoldi Algorithm for Single-Output Sylvester-Observer Equation Arnoldi Algorithm for CARE Projection Algorithm for Partial Pole-Placement Lanczos Algorithm for SISO Model Reduction Amoldi Algorithm for SISO Model Reduction
651 652 653
15.4.3 15.4.4 15.4.5 15.4.6 15.5.1 15.6.1 15.6.2
654 655 656 657 658 659 660 661