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AB08NXConstruction of a reduced system with input/output matrix Dr of full row rank, preserving transmission zerosAB08NYConstruction of a reduced system with input/output matrix Dr of full row rank, preserving transmission zeros (extended variant)AB8NXZConstruction of a reduced system with input/output matrix Dr of full row rank, preserving transmission zeros (complex case)

AB09AXBalance & Truncate model reduction with state matrix in real Schur formAB09BXSingular perturbation approximation based model reduction with state matrix in real Schur formAB09CXHankel norm approximation based model reduction with state matrix in real Schur formAB09HXStochastic balancing model reduction of stable systemsAB09HYCholesky factors of the controllability and observability GrammiansAB09IXAccuracy enhanced balancing related model reductionAB09IYCholesky factors of the frequency-weighted controllability and observability GrammiansAB09JVState-space representation of a projection of a left weighted transfer-function matrixAB09JWState-space representation of a projection of a right weighted transfer-function matrixAB09JXCheck stability/antistability of finite eigenvaluesAB09KXStable projection of V*G*W or conj(V)*G*conj(W)

AB13AXHankel-norm of a stable system with state matrix in real Schur formAB13DXMaximum singular value of a transfer-function matrix

AG08BYConstruction of a reduced system with input/output matrix Dr of full row rank, preserving the finite Smith zerosAG8BYZConstruction of a reduced system with input/output matrix Dr of full row rank, preserving the finite Smith zeros (complex case)

IB01MDUpper triangular factor in QR factorization of a block-Hankel-block matrixIB01MYUpper triangular factor in fast QR factorization of a block-Hankel-block matrixIB01NDSingular value decomposition giving the system orderIB01ODEstimating the system orderIB01OYUser's confirmation of the system orderIB01PDEstimating the system matrices and covariancesIB01PXEstimating the matrices B and D of a system using Kronecker productsIB01PYEstimating the matrices B and D of a system exploiting the structureIB01QDEstimating the initial state and the matrices B and D of a systemIB01RDEstimating the initial state of a system

MA01ADComplex square root of a complex number in real arithmeticMA01BDSafely computing the general product of K real scalarsMA01BZSafely computing the general product of K complex scalarsMA01CDSafely computing the sign of a sum of two real numbers represented using integer powers of a base

MA02ADTranspose of a matrixMA02BDReversing the order of rows and/or columns of a matrixMA02BZReversing the order of rows and/or columns of a matrix (complex case)MA02CDPertranspose of the central band of a square matrixMA02CZPertranspose of the central band of a square matrix (complex case)MA02DDPack/unpack the upper or lower triangle of a symmetric matrixMA02EDConstruct a triangle of a symmetric matrix, given the other triangleMA02ESConstruct a triangle of a skew-symmetric real matrix, given the other triangleMA02EZConstruct a triangle of a (skew-)symmetric/Hermitian complex matrix, given the other triangleMA02FDHyperbolic plane rotationMA02GDColumn interchanges on a real matrixMA02GZColumn interchanges on a complex matrixMA02HDCheck if a matrix is a scalar multiple of an identity-like matrixMA02HZCheck if a complex matrix is a scalar multiple of an identity-like matrixMA02IDMatrix 1-, Frobenius, or infinity norms of a skew-Hamiltonian matrixMA02IZMatrix 1-, Frobenius, or infinity norms of a complex skew-Hamiltonian matrixMA02JDTest if a matrix is an orthogonal symplectic matrixMA02JZTest if a matrix is a unitary symplectic matrixMA02MDNorms of a real skew-symmetric matrixMA02MZNorms of a complex skew-symmetric matrixMA02NZTwo rows and columns permutation of a (skew-)symmetric/Hermitian complex matrixMA02ODNumber of zero rows of a real (skew-)Hamiltonian matrixMA02OZNumber of zero rows of a complex (skew-)Hamiltonian matrixMA02PDNumber of zero rows and columns of a real matrixMA02PZNumber of zero rows and columns of a complex matrixMB01KDRank 2k operation alpha*A*trans(B) - alpha*B*trans(A) + beta*C, with A and C skew-symmetric matricesMB01LDComputation of matrix expression alpha*R + beta*A*X*trans(A) with skew-symmetric matrices R and XMB01MDMatrix-vector operation alpha*A*x + beta*y, with A a skew-symmetric matrixMB01NDRank 2 operation alpha*x*trans(y) - alpha*y*trans(x) + A, with A a skew-symmetric matrixMB01SDRows and/or columns scaling of a matrix

MB01RUComputation of matrix expression alpha*R + beta*A*X*trans(A) (MB01RD variant)MB01RWComputation of matrix expression alpha*A*X*trans(A), X symmetric (BLAS 2)MB01RXComputing a triangle of the matrix expressions alpha*R + beta*A*B or alpha*R + beta*B*AMB01RYComputing a triangle of the matrix expressions alpha*R + beta*H*B or alpha*R + beta*B*H, with H an upper Hessenberg matrixMB01UWComputation of matrix expressions alpha*H*A or alpha*A*H, overwritting A, with H an upper Hessenberg matrixMB01VDKronecker product of two matricesMB01XYComputation of the product U'*U or L*L', with U and L upper and lower triangular matrices (unblock algorithm)SB03OVConstruction of a complex plane rotation to annihilate a real number, modifying a complex numberSG03BYComputing a complex plane rotation in real arithmetic

MB02CUBringing the first blocks of a generator in proper form (extended version of MB02CX)MB02CVApplying the MB02CU transformations on other columns / rows of the generatorMB02CXBringing the first blocks of a generator in proper formMB02CYApplying the MB02CX transformations on other columns / rows of the generatorMB02NYSeparation of a zero singular value of a bidiagonal submatrixMB02QYMinimum-norm least squares solution, given a rank-revealing QR factorizationMB02UUSolution of linear equations using LU factorization with complete pivotingMB02UVLU factorization with complete pivotingMB02UWSolution of linear equations of order at most 2 with possible scaling and perturbation of system matrixMB02WDSolution of a positive definite linear system A*x = b, or f(A, x) = b, using conjugate gradient algorithmMB02XDSolution of a set of positive definite linear systems, A'*A*X = B, or f(A)*X = B, using Gaussian eliminationMB02YDSolution of the linear system A*x = b, D*x = 0, D diagonal

MB03ADReducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vectorMB03BAComputing maps for Hessenberg index and signature arrayMB03BBEigenvalues of a 2-by-2 matrix product via a complex single shifted periodic QZ algorithmMB03BCProduct singular value decomposition of K-1 triangular factors of order 2MB03BDFinding eigenvalues of a generalized matrix product in Hessenberg-triangular formMB03BEApplying iterations of a real single shifted periodic QZ algorithm to a 2-by-2 matrix productMB03BZFinding eigenvalues of a complex generalized matrix product in Hessenberg-triangular formMB03CDExchanging eigenvalues of a real 2-by-2, 3-by-3 or 4-by-4 block upper triangular pencil (factored version)MB03CZExchanging eigenvalues of a complex 2-by-2 upper triangular pencil (factored version)MB03DDExchanging eigenvalues of a real 2-by-2, 3-by-3 or 4-by-4 block upper triangular pencilMB03DZExchanging eigenvalues of a complex 2-by-2 upper triangular pencilMB03EDReducing a real 2-by-2 or 4-by-4 block (anti-)diagonal skew-Hamiltonian/Hamiltonian pencil to generalized Schur form and moving eigenvalues with negative real parts to the top (factored version)MB03FDReducing a real 2-by-2 or 4-by-4 block (anti-)diagonal skew-Hamiltonian/Hamiltonian pencil to generalized Schur form and moving eigenvalues with negative real parts to the topMB03GDExchanging eigenvalues of a real 2-by-2 or 4-by-4 block upper triangular skew-Hamiltonian/Hamiltonian pencil (factored version)MB03GZExchanging eigenvalues of a complex 2-by-2 skew-Hamiltonian/ Hamiltonian pencil in structured Schur form (factored version)MB03HDExchanging eigenvalues of a real 2-by-2 or 4-by-4 skew-Hamiltonian/ Hamiltonian pencil in structured Schur formMB03HZExchanging eigenvalues of a complex 2-by-2 skew-Hamiltonian/ Hamiltonian pencil in structured Schur formMB03IDMoving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (factored version)MB03IZMoving eigenvalues with negative real parts of a complex skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (factored version)MB03JDMoving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencilMB03JPMoving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (applying transformations on panels of columns)MB03JZMoving eigenvalues with negative real parts of a complex skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencilMB3JZPMoving eigenvalues with negative real parts of a complex skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (applying transformations on panels of columns)MB03KAMoving diagonal blocks at a specified position in a formal matrix product to another positionMB03KBSwapping pairs of adjacent diagonal blocks of sizes 1 and/or 2 in a formal matrix productMB03KCReducing a 2-by-2 formal matrix product to periodic Hessenberg-triangular formMB03KDReordering the diagonal blocks of a formal matrix product using periodic QZ algorithmMB03KESolving periodic Sylvester-like equations with matrices of order at most 2MB03NYThe smallest singular value of A - jwIMB03OYMatrix rank determination by incremental condition estimation, during the pivoted QR factorization processMB3OYZMatrix rank determination by incremental condition estimation, during the pivoted QR factorization process (complex case)MB03PYMatrix rank determination by incremental condition estimation, during the pivoted RQ factorization process (row pivoting)MB3PYZMatrix rank determination by incremental condition estimation, during the pivoted RQ factorization process (row pivoting, complex case)MB03QVEigenvalues of an upper quasi-triangular matrix pencilMB03QWStandardization and eigenvalues of a 2-by-2 diagonal block pair of an upper quasi-triangular matrix pencilMB03QXEigenvalues of an upper quasi-triangular matrixMB03QYTransformation to Schur canonical form of a selected 2-by-2 diagonal block of an upper quasi-triangular matrixMB03RXReordering the diagonal blocks of a principal submatrix of a real Schur form matrixMB03RYTentative solution of Sylvester equation -AX + XB = C (A, B in real Schur form)MB03TSSwapping two diagonal blocks of a matrix in (skew-)Hamiltonian canonical Schur formMB03VYGenerating orthogonal matrices for reduction to periodic Hessenberg form of a product of matricesMB03WASwapping two adjacent diagonal blocks in a periodic real Schur canonical formMB03WXEigenvalues of a product of matrices, T = T_1*T_2*...*T_p, with T_1 upper quasi-triangular and T_2, ..., T_p upper triangularMB03XSEigenvalues and real skew-Hamiltonian Schur form of a skew-Hamiltonian matrixMB03XUPanel reduction of columns and rows of a real (k+2n)-by-(k+2n) matrix by orthogonal symplectic transformationsMB03YAAnnihilation of one or two entries on the subdiagonal of a Hessenberg matrix corresponding to zero elements on the diagonal of a triangular matrixMB03YTPeriodic Schur factorization of a real 2-by-2 matrix pair (A,B) with B upper triangularMB03ZAReordering a selected cluster of eigenvalues of a given matrix pair in periodic Schur formMB05MYComputing an orthogonal matrix reducing a matrix to real Schur form T, the eigenvalues, and the upper triangular matrix of right eigenvectors of TMB05OYRestoring a matrix after balancing transformations

MB04CDReducing a special real block (anti-)diagonal skew-Hamiltonian/ Hamiltonian pencil in factored form to generalized Schur formMB04DBApplying the inverse of a balancing transformation for a real skew-Hamiltonian/Hamiltonian matrix pencilMB4DBZApplying the inverse of a balancing transformation for a complex skew-Hamiltonian/Hamiltonian matrix pencilMB04DDBalancing a real Hamiltonian matrixMB04DZBalancing a complex Hamiltonian matrixMB04DIApplying the inverse of a balancing transformation for a real Hamiltonian matrixMB04DSBalancing a real skew-Hamiltonian matrixMB04DYSymplectic scaling of a Hamiltonian matrixMB04HDReducing a special real block (anti-)diagonal skew-Hamiltonian/ Hamiltonian pencil to generalized Schur formMB04IYApplying the product of elementary reflectors used for QR factorization of a matrix having a lower left zero triangleMB04NYApplying an elementary reflector to a matrix C = ( A B ), from the right, where A has one columnMB04OYApplying an elementary reflector to a matrix C = ( A' B' )', from the left, where A has one rowMB04OWRank-one update of a Cholesky factorization for a 2-by-2 block matrixMB04OXRank-one update of a Cholesky factorizationMB04PUComputation of the Paige/Van Loan (PVL) form of a Hamiltonian matrix (unblocked algorithm)MB04PYApplying an elementary reflector to a matrix from the left or rightMB04QBApplying a product of symplectic reflectors and Givens rotations to two general real matricesMB04QCPremultiplying a real matrix with an orthogonal symplectic block reflectorMB04QFForming the triangular block factors of a symplectic block reflectorMB04QSMultiplication with a product of symplectic reflectors and Givens rotationsMB04QUApplying a product of symplectic reflectors and Givens rotations to two general real matrices (unblocked algorithm)MB04RBReduction of a skew-Hamiltonian matrix to Paige/Van Loan (PVL) form (blocked version)MB04RUReduction of a skew-Hamiltonian matrix to Paige/Van Loan (PVL) form (unblocked version)MB04SUSymplectic QR decomposition of a real 2M-by-N matrixMB04TSSymplectic URV decomposition of a real 2N-by-2N matrix (unblocked version)MB04TUApplying a row-permuted Givens transformation to two row vectorsMB04WDGenerating an orthogonal basis spanning an isotropic subspaceMB04WPGenerating an orthogonal symplectic matrix which performed the reduction in MB04PUMB04WRGenerating orthogonal symplectic matrices defined as products of symplectic reflectors and Givens rotationsMB04WUGenerating an orthogonal basis spanning an isotropic subspace (unblocked version)MB04XYApplying Householder transformations for bidiagonalization (stored in factored form) to one or two matrices, from the leftMB04YWOne QR or QL iteration step onto an unreduced bidiagonal submatrix of a bidiagonal matrix

MC01PYCoefficients of a real polynomial, stored in decreasing order, given its zeros

MC03NXConstruction of a pencil sE-A related to a given polynomial matrix

MD03BXQR factorization with column pivoting and error vector transformationMD03BYFinding the Levenberg-Marquardt parameter

NF01ADComputing the output of a Wiener systemNF01AYComputing the output of a set of neural networksNF01BDComputing the Jacobian of a Wiener systemNF01BPFinding the Levenberg-Marquardt parameterNF01BQSolution of the linear system J*x = b, D*x = 0, D diagonalNF01BRSolution of the linear system op(R)*x = b, R block upper triangular stored in a compressed formNF01BSQR factorization of a structured Jacobian matrixNF01BUComputing J'*J + c*I, for the Jacobian J given in a compressed formNF01BVComputing J'*J + c*I, for a full Jacobian J (one output variable)NF01BWMatrix-vector product x <-- (J'*J + c*I)*x, for J in a compressed formNF01BXMatrix-vector product x <-- (A'*A + c*I)*x, for a full matrix ANF01BYComputing the Jacobian of the error function for a neural network (for one output variable)

SB01BXChoosing the closest real (complex conjugate) eigenvalue(s) to a given real (complex) valueSB01BYPole placement for systems of order 1 or 2SB01FYInner denominator of a right-coprime factorization of an unstable system of order 1 or 2

SB02MUConstructing the 2n-by-2n Hamiltonian or symplectic matrix for linear-quadratic optimization problemsSB02RUConstructing the 2n-by-2n Hamiltonian or symplectic matrix for linear-quadratic optimization problems (efficient and accurate version of SB02MU)SB02OYConstructing and compressing the extended Hamiltonian or symplectic matrix pairs for linear-quadratic optimization problems

SB03MVSolving a discrete-time Lyapunov equation for a 2-by-2 matrixSB03MWSolving a continuous-time Lyapunov equation for a 2-by-2 matrixSB03MXSolving a discrete-time Lyapunov equation with matrix A quasi-triangularSB03MYSolving a continuous-time Lyapunov equation with matrix A quasi-triangularSB03OTSolving (for Cholesky factor) stable continuous- or discrete-time Lyapunov equations, with A quasi-triangular and R triangularSB03OUSolving (for Cholesky factor) stable continuous- or discrete-time Lyapunov equations, with A in real Schur form and B rectangularSB03OYSolving (for Cholesky factor) stable 2-by-2 continuous- or discrete-time Lyapunov equations, with matrix A having complex conjugate eigenvaluesSB03QXForward error bound for continuous-time Lyapunov equationsSB03QYSeparation and Theta norm for continuous-time Lyapunov equationsSB03SXForward error bound for discrete-time Lyapunov equationsSB03SYSeparation and Theta norm for discrete-time Lyapunov equations

SB03MUSolving a discrete-time Sylvester equation for an m-by-n matrix X, 1 <= m,n <= 2SB03ORSolving quasi-triangular continuous- or discrete-time Sylvester equations, for an n-by-m matrix X, 1 <= m <= 2SB04MRSolving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the second subdiagonalSB04MUConstructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the second subdiagonalSB04MWSolving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the first subdiagonalSB04MYConstructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the first subdiagonalSB04NVConstructing right-hand sides for a system of equations in Hessenberg form solved via SB04NXSB04NWConstructing the right-hand side for a system of equations in Hessenberg form solved via SB04NYSB04NXSolving a system of equations in Hessenberg form with two consecutive offdiagonals and two right-hand sidesSB04NYSolving a system of equations in Hessenberg form with one offdiagonal and one right-hand sideSB04OWSolving a periodic Sylvester equation with matrices in periodic Schur formSB04PXSolving a discrete-time Sylvester equation for matrices of order <= 2SB04PYSolving a discrete-time Sylvester equation with matrices in Schur formSB04QRSolving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the third subdiagonalSB04QUConstructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the third subdiagonalSB04QYConstructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the first subdiagonal (discrete-time case)SB04RVConstructing right-hand sides for a system of equations in Hessenberg form solved via SB04RXSB04RWConstructing the right-hand side for a system of equations in Hessenberg form solved via SB04RYSB04RXSolving a system of equations in Hessenberg form with two consecutive offdiagonals and two right-hand sides (discrete-time case)SB04RYSolving a system of equations in Hessenberg form with one offdiagonal and one right-hand side (discrete-time case)

SB10JDConversion of a descriptor state-space system into regular state-space formSB10LDClosed-loop system matrices for a system with robust controllerSB10PDNormalization of a system for H-infinity controller designSB10QDState feedback and output injection matrices for an H-infinity (sub)optimal state controller (continuous-time)SB10RDH-infinity (sub)optimal controller matrices using state feedback and output injection matrices (continuous-time)SB10SDH2 optimal controller matrices for a normalized discrete-time systemSB10TDH2 optimal controller matrices for a discrete-time systemSB10UDNormalization of a system for H2 controller designSB10VDState feedback and output injection matrices for an H2 optimal state controller (continuous-time)SB10WDH2 optimal controller matrices using state feedback and output injection matrices (continuous-time)SB10YDFitting frequency response data with a stable, minimum phase SISO systemSB10ZPTransforming a SISO system into a stable and minimum phase one

SB16AYCholesky factors of the frequency-weighted controllability and observability Grammians for controller reductionSB16CYCholesky factors of controllability and observability Grammians of coprime factors of a state-feedback controller

SG03AXSolving a generalized discrete-time Lyapunov equation with A quasi-triangular and E upper triangularSG03AYSolving a generalized continuous-time Lyapunov equation with A quasi-triangular and E upper triangularSG03BUSolving (for Cholesky factor) stable generalized discrete-time Lyapunov equations with A quasi-triangular, and E, B upper triangularSG03BVSolving (for Cholesky factor) stable generalized continuous-time Lyapunov equations with A quasi-triangular, and E, B upper triangularSG03BXSolving (for Cholesky factor) stable generalized 2-by-2 Lyapunov equations

SG03BWSolving a generalized Sylvester equation with A quasi-triangular and E upper triangular, for X m-by-n, n = 1 or 2

TB01KXAdditive spectral decomposition of the transfer-function matrix of a standard systemTB01UXObservable-unobservable decomposition of a standard systemTB01VDConversion of a discrete-time system to output normal formTB01VYConversion of the output normal form of a discrete-time system to a state-space representationTB01XDSpecial similarity transformation of the dual state-space systemTB01XZSpecial similarity transformation of the dual state-space system (complex case)TB01YDSpecial similarity transformation of a state-space system

TB04BVStrictly proper part of a proper transfer function matrixTB04BWSum of a rational matrix and a real matrixTB04BXGain of a SISO linear system, given (A,b,c,d), its poles and zeros

TF01MXOutput response of a linear discrete-time system, given a general system matrix (each output is a column of the result)TF01MYOutput response of a linear discrete-time system, given the system matrices (each output is a column of the result)

TG01HUStaircase controllability representation of a multi-input descriptor systemTG01HXOrthogonal reduction of a descriptor system to a system with the same transfer-function matrix and without uncontrollable finite eigenvaluesTG01HYOrthogonal reduction of a descriptor system to a system with the same transfer-function matrix and without uncontrollable finite eigenvalues (blocked version)TG01LYFinite-infinite decomposition of a structured descriptor systemTG01NXBlock-diagonal decomposition of a descriptor system in generalized real Schur form