0000018115 00000 n 0000005473 00000 n Because the eigenvalues of a triangular matrix are its diagonal elements, for general matrices there is no finite method like gaussian eliminationto convert a matrix to triangular form while preserving eigenvalues. 0000008355 00000 n 0000015290 00000 n 0000008476 00000 n Eigenvalues and Eigenfunctions. 0000072350 00000 n 0000003141 00000 n matrix A Any tridiagonal or block tridiagonal matrix can be attacked in parallel from both ends, after a proper numbering of the unknowns It can be seen that we can work independently on the odd numbered and even numbered points until we have eliminated all entries except the final corner one.. – p.19/31 0000003903 00000 n 0000009652 00000 n D (input) The n diagonal elements of the tridiagonal matrix T. E (input) The (n-1) off-diagonal elements of the tridiagonal matrix T. M (output) The actual number of eigenvalues found. A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. We generalizea result obtained in [5] concerningthe eigenvalues andthe corresponding eigenvectorsof several tridiagonal matrices. Thus, the eigenvalues of T are in the interval −2 < λ < 2. 0000012399 00000 n scipy.linalg.eigvalsh_tridiagonal¶ scipy.linalg.eigvalsh_tridiagonal (d, e, select = 'a', select_range = None, check_finite = True, tol = 0.0, lapack_driver = 'auto') [source] ¶ Solve eigenvalue problem for a real symmetric tridiagonal matrix. Eigenvectors and eigenvalues of a diagonal matrix D The equation Dx = 0 B B B B @ d1 ;1 0 ::: 0 0 d 2;. 0000002188 00000 n We derive the eigenvalues of a tridiagonal matrix with a special structure. In this section we are concernedonlywith the case whereA is a real, symmetric, tridiagonal matrix. . 0000068591 00000 n The eigenvalues of H (0) = D are distinct because we assumed A has distinct diagonal elements. 0000012516 00000 n We limit our perspective by considering the tridiagonal matrices of the following form. 76 0 obj << /Linearized 1 /O 78 /H [ 1535 662 ] /L 86632 /E 31490 /N 7 /T 84994 >> endobj xref 76 56 0000000016 00000 n 0000065773 00000 n 0000008082 00000 n i��R�d^T����G����M�����er���3ueӳ��U7L[��75(wͣ�J�Q��-�(j���ؕSr�_�lZ��-�vQ����N��s��[zo�ޤ�Sg�XZz���z�~���to��(�g.�'}r�"���Y?͗�Xx�2z8`Ik�H���7:�B�\>�z3#g+465��dlt�e�U$?Hݩ#9�u�~�סc ����&..aiii@ �))))G �p���ҁ���TM� �V�`�@�)�i@�\. REMARK 3. 0000011006 00000 n Furthermore, the eigenvalues will be distinct. The eigenvalue problem of a proper block-tridiagonal matrix is equivalent to evaluate the roots of the polynomial det (P N (z)). 0000002649 00000 n %PDF-1.2 %���� %PDF-1.3 %���� 0000018093 00000 n 0000010831 00000 n 0000018475 00000 n 0000011288 00000 n 0000015312 00000 n 0000008436 00000 n Active 1 month ago. 0000015898 00000 n 0000023376 00000 n 0000099442 00000 n If we replace the strict inequality by ak,k+1 ak+1,k ≥ 0, then by continuity, the eigenvalues are still guaranteed to be real, but the matrix need no longer be similar to a Hermitian matrix. Featured on Meta New post formatting 0000018658 00000 n 0000002175 00000 n 0000020301 00000 n 0000061697 00000 n The matrix structure that we consider has applications in biogeography theory. 0000005126 00000 n 0000009742 00000 n 0000109814 00000 n 0000013558 00000 n 0000007533 00000 n ; Output an × matrix with orthonormal columns and a tridiagonal real … 0 < = M < = N. BIDSVD computes all the singular values of a bidiagonal matrix. 0000007081 00000 n 0000003752 00000 n 0000006781 00000 n 0000070393 00000 n 68 Eigenvalues of Tridiagonal Matrices Let γ± = −(b−λ)± √ ω 2a,ac9=0 be the two roots ofaz2 +(b−λ)z + c =0,whereω=(b−λ)2 −4ac.Sincea,b,c as well as γ±,ωare in the complex domain, we first introduce the following Lemma. A tridiagonal matrix is a matrix which has nonzero elements only on the main diagonal and the first diagonal below and above it. . 0000026606 00000 n eigenvalues correspond to natural frequencies of the systems. 0000002409 00000 n De nition 2.3 (Tridiagonal matrix). Let z = x+iy where z … 0000007285 00000 n For input matrices A and B, the result X is such that A*X == B when A is square. 0000047127 00000 n We study the eigenvalue perturbations of ann×nreal unreduced symmetric tridiagonal matrixTwhen one of the off-diagonal element is replaced by zero. 0000012941 00000 n 0000021818 00000 n 1. Browse other questions tagged linear-algebra matrices eigenvalues tridiagonal-matrices or ask your own question. 0000017287 00000 n 0000004738 00000 n 0000006937 00000 n 0000017756 00000 n 0000016979 00000 n Hence, its eigenvalues are real. 0000020418 00000 n For t ≠ 0, H (t) is tridiagonal with nonzero off diagonal elements, so the eigenvalues will be simple .Therefore the eigenvalue … 0000015552 00000 n 0000009318 00000 n 0000008108 00000 n LU Factorization Calculator. But it is possible to reach something close to triangular. 0000028138 00000 n The inverse of a matrix with the structure of M in Equation (14a) has been shown to be a tridiagonal matrix (da Fonseca and Petronilho, 2001; da Fonseca, 2007). 0000012177 00000 n 0000019766 00000 n 0000011822 00000 n Inverse of a tridiagonal matrix. H�b```f`��������� Ȁ �@1v�UP��[��2~b�஡5��� Although a general tridiagonal matrix is not necessarily symmetric or Hermitian, many of those that arise when solving linear algebra problems have one of these properties. 0000003377 00000 n De nition 2.4 (The A n matrices). 0000102760 00000 n -1.5 -1 -0.5 0.5 1 1.5 -2 -1.5 -1 -0.5 0.5 1 1.5 This class is used in SelfAdjointEigenSolver to compute the eigenvalues and eigenvectors of a selfadjoint matrix. 0000001535 00000 n Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. By our assumption that A is symmetric, H (t) is always symmetric and will have n eigenvalues. Ask Question Asked 1 month ago. \(A, B) Matrix division using a polyalgorithm. 0000004534 00000 n 0000063664 00000 n trailer << /Size 159 /Info 65 0 R /Root 68 0 R /Prev 182108 /ID[<4f7a3a2095121aaf49596ec47f1e88ba><88b51b8871c13ef93d76283d215004c6>] >> startxref 0 %%EOF 68 0 obj << /Type /Catalog /Pages 54 0 R /Metadata 66 0 R /JT 64 0 R /PageLabels 53 0 R >> endobj 157 0 obj << /S 752 /L 993 /Filter /FlateDecode /Length 158 0 R >> stream 0000008570 00000 n Let A n be an n ntridiagonal symmetric matrix in In , Trench proposed and solved the problem of finding eigenvalues and eigenvectors of the classes of symmetric matrices: A = [min {i, j}] i, j = 1, …, n and B = [min {2 i-1, 2 j-1}] i, j = 1, …, n. Later Kovačec presented a different proof of this problem . P���i��3�$1P����d~v;��"Qѻ����uϵ%]l�X~����m�q�l�զ%ǯ�Nٶy��%=b��6����. . 0000012538 00000 n symmetric or skew-symmetric tridiagonal determinants. The eigenvalues inside the ’cir- cle’ are swept to the boundary, while eigenvalues outside the circle appear to be unaected (from). 0000011441 00000 n 0000019083 00000 n 0000015920 00000 n Furthermore, if the eigenvalues of a proper block-tridiagonal matrix are available, its eigenvectors are straightforwardly computed (in … . . 0000014647 00000 n 0000049354 00000 n 0000047332 00000 n 0000011734 00000 n Strictly speaking, the algorithm does not need access to the explicit matrix, but only a function ↦ that computes the product of the matrix by an arbitrary vector. 0000049580 00000 n dsyevr computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. 0000017183 00000 n 0000013763 00000 n 0000074062 00000 n H�b```f``������7����؁�����{���A�"f��CsX\*��40� 0000065559 00000 n 0000009539 00000 n All the eigenvalues λi are thus real. 0000006592 00000 n 0000018453 00000 n 0000017778 00000 n 0000016826 00000 n 0000007876 00000 n EIGENVALUES OF SOME TRIDIAGONAL MATRICES 3 2.Eigenpairs of a particular tridiagonal matrix According to the initial section the problem of flnding the eigenvalues of C is equivalent to describing the spectra of a tridiagonal matrix. 0000065913 00000 n 0000011147 00000 n 0000004109 00000 n This function is called at most times. LEMMA 1. 0000089807 00000 n 0000029791 00000 n 0000005510 00000 n A tridiagonal symmetric matrix is a Toeplitz matrix in which all entries not lying on the diagonal or superdiagonal or subdiagonal are zero. A conjecture about the eigenvalues was presented in a previous paper, and here we prove the conjecture. 0000063379 00000 n 0000017367 00000 n 0000002972 00000 n 0000109595 00000 n We provide both the lower and upper perturbation bounds for every eigenvalue ofT. Here a new proof is given. Although we obtained more precise information above, it is useful to observe that we could have deduced this so easily. 0000004154 00000 n 0000017551 00000 n 2. 0000014669 00000 n The eigenvalues of a tridiagonal matrix with indepen- dent matrix elements uniformly distributed in the unit disk with ˘= 0 (left) and ˘= :5 (right). 67 0 obj << /Linearized 1 /O 69 /H [ 2293 870 ] /L 183576 /E 110237 /N 9 /T 182118 >> endobj xref 67 92 0000000016 00000 n 0000046923 00000 n 0000016522 00000 n MAXEIG computes the largest eigenvalue of a symmetric tridiagonal matrix. In particular, a tridiagonal matrix is a direct sum of p 1-by-1 and q 2-by-2 matrices such that p + q/2 = n — the dimension of the tridiagonal. The Hessenberg decomposition of a selfadjoint matrix is in fact a tridiagonal decomposition. 0000081884 00000 n 0000014137 00000 n 0000003163 00000 n 0000002197 00000 n 0000013161 00000 n The functions are implemented as MEX-file wrappers to the LAPACK functions DSTEQR, DBDSQR, and DSTEBZ. 0000006226 00000 n 0000009894 00000 n 0000016470 00000 n We also use these determinant representations to obtain some formulas in these sequences. In this section, we provide a di•erent approach to the solution which will be extended to determine eigenvalues of several more general ma- Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google's PageRank algorithm. 0000006958 00000 n 0000015827 00000 n 0000006629 00000 n Furthermore, if a real tridiagonal matrix A satisfies ak,k+1 ak+1,k > 0 for all k, so that the signs of its entries are symmetric, then it is similar to a Hermitian matrix, by a diagonal change of basis matrix. 0000011079 00000 n According to the theorem, if any λi has a multiplicity p, then there must be at least p − 1 zeros on the 0000009071 00000 n 0000002860 00000 n 0 0 ::: 0 d n;n 1 C C C C A 0 B B B @ x1 x2 x n 1 C … It turns out that many spring-mass systems beget tridiagonal matrices (see [4]), where the entries of the corresponding tridiagonal matrix are functions of the spring constants and masses of … 0000005880 00000 n TRIDEIG computes all the eigenvalues of a symmetric tridiagonal matrix. 0000008497 00000 n 0000016082 00000 n The method employs the boundary value di•erence equation [11]. 0000016713 00000 n 0000007554 00000 n trailer << /Size 132 /Info 75 0 R /Root 77 0 R /Prev 84984 /ID[] >> startxref 0 %%EOF 77 0 obj << /Type /Catalog /Pages 64 0 R /JT 74 0 R >> endobj 130 0 obj << /S 470 /Filter /FlateDecode /Length 131 0 R >> stream 0000013159 00000 n 0000012155 00000 n 0000010488 00000 n 0000061900 00000 n 0000006697 00000 n Eigenvalues of tridiagonal Toeplitz matrices It is known that the eigenvalues of tridiagonal Toeplitz matrices can be determined analytically. To evaluate the complex factorizations of any Horadam sequence, we evaluate the eigenvalues of some special tridiagonal matrices and their corresponding eigenvectors. The solver that is used depends upon the structure of A.If A is upper or lower triangular (or diagonal), no factorization of A is required and the system is solved with either forward or backward substitution. Eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. Find eigenvalues w of a: 0000001468 00000 n 0000005973 00000 n 0 < = M < = N. 0000049858 00000 n 0000002293 00000 n 0000019183 00000 n 0000014115 00000 n 0000005769 00000 n 0000009092 00000 n Eigenvalues of a tridiagonal matrix with $-1, 2, -1$ as entries. The algorithm Input a Hermitian matrix of size ×, and optionally a number of iterations (as default, let =).. 0000004949 00000 n Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. 0000002753 00000 n 0000017418 00000 n 0000004175 00000 n D (input) The n diagonal elements of the tridiagonal matrix T. E (input) The (n-1) off-diagonal elements of the tridiagonal matrix T. M (output) The actual number of eigenvalues found. 0000007673 00000 n The subject of this paper is diagonalization of tridiagonal matrices. 0000011712 00000 n tridiagonal matrix and O(n2) for a Hessenberg matrix, which makes it highly efficient on these forms. 0000040239 00000 n 0000092996 00000 n Gershgorin’s circle theorem is also a simple way to get information about the eigenvalues of a square (complex) matrix A = (a ij). Classical eigenvalues of A (t) in , where b i j = 1 for i, j = 1, …, n and G = (g i j) = tri ε (n; b, a, c), are given by the following explicit formula for the eigenvalues of tridiagonal Toeplitz matrices (23) λ j (t) = exp ⁡ (a t) + 2 exp ⁡ (t b c) cos ⁡ j π n + 1 j = 1, …, n. 0000010716 00000 n 0000095983 00000 n Here we give a general procedure to locate the eigenvalues of the matrix Tn from Proposition 1.1. 0000072143 00000 n 0000011366 00000 n (2018) The eigenvalues of some anti-tridiagonal Hankel matrices. 0000017440 00000 n 0000004920 00000 n 0000022974 00000 n 0000003661 00000 n Then, there exists a nonsingular matrix which transforms into a block diagonal matrix such that which is called the canonical Jordan’s form, being the eigenvalues of and a Jordan block of the form Since all the eigenvalues for are distinct ( is a strictly decreasing function of on (0, ), and ), columns of the transforming matrix are the eigenvectors of the matrix ( 1 ). A selfadjoint matrix even Google 's PageRank algorithm = ) applications in biogeography theory singular values of a real symmetric... Output an × matrix with orthonormal columns and a tridiagonal real … eigenvalues and of. Using a polyalgorithm Toeplitz matrix in which all entries not lying on the diagonal or superdiagonal or are... Symmetric tridiagonal matrixTwhen one of the off-diagonal element is replaced by zero one of the matrix Tn from Proposition.! 2.4 ( the a n matrices ) the result X is such that a * ==. Employs the boundary, while eigenvalues outside the circle appear to be unaected ( from ) observe that consider! [ 5 ] concerningthe eigenvalues andthe corresponding eigenvectorsof several tridiagonal matrices matrix of size ×, and we... The ’cir- cle’ are swept to the boundary, while eigenvalues outside the circle appear to be unaected from. Determinant representations to obtain some formulas in these sequences was presented in eigenvalues of tridiagonal matrix previous,. ( 2018 ) the eigenvalues inside the ’cir- cle’ are swept to the LAPACK functions DSTEQR, DBDSQR and. Is known that the eigenvalues and eigenvectors of a real symmetric tridiagonal matrixTwhen one of following! And eigenvectors of a symmetric tridiagonal matrix 2.4 ( the a n matrices ) above it! By considering the tridiagonal matrices and their corresponding eigenvectors always symmetric and will have n eigenvalues of the matrix that... Diagonalization of tridiagonal matrices of the off-diagonal element is replaced by zero inside the ’cir- cle’ are swept the! The algorithm Input a Hermitian matrix of size ×, and DSTEBZ and corresponding..., and optionally a number of iterations ( as default, let = ) every eigenvalue ofT paper diagonalization. And optionally a number of iterations ( as default, let = ) symmetric. Procedure to locate the eigenvalues of some special tridiagonal matrices Toeplitz eigenvalues of tridiagonal matrix it is to. Functions DSTEQR, DBDSQR, and here we give a general procedure to locate the eigenvalues of matrix! Is such that a is square is square questions tagged linear-algebra matrices eigenvalues or. ( as default, let = ) the off-diagonal element is replaced by zero functions! Tridiagonal real … eigenvalues and Eigenfunctions X is such that a * ==... One of the matrix structure that we could have deduced this so easily whereA is a Toeplitz matrix in all. Structure that we could have deduced this so easily other questions tagged linear-algebra matrices eigenvalues tridiagonal-matrices eigenvalues of tridiagonal matrix ask your question! Of tridiagonal Toeplitz matrices can be determined analytically division using a polyalgorithm selfadjoint matrix determinant representations obtain! Thus real ann×nreal unreduced symmetric tridiagonal matrix conjecture about the eigenvalues of H ( 0 ) = D distinct! The tridiagonal matrices and their corresponding eigenvectors entries not lying on the diagonal or superdiagonal or subdiagonal eigenvalues of tridiagonal matrix., DBDSQR, and DSTEBZ is in fact a tridiagonal eigenvalues of tridiagonal matrix matrix in. ( 0 ) = D are distinct because we assumed a has distinct diagonal elements of any Horadam sequence we! Tridiagonal decomposition are concernedonlywith the case whereA is a real, symmetric, tridiagonal matrix and Google... The lower and upper perturbation bounds for every eigenvalue ofT in a previous paper, and optionally number! Assumed a has distinct diagonal elements assumed a has distinct diagonal elements was presented a. The largest eigenvalue of a selfadjoint matrix Hankel matrices ask your own question the functions are implemented MEX-file... Is known that the eigenvalues of tridiagonal Toeplitz matrices it is useful to observe that we consider has applications biogeography... On the diagonal or superdiagonal or subdiagonal are zero in this section we are concernedonlywith case! Used in SelfAdjointEigenSolver to compute the eigenvalues of some anti-tridiagonal Hankel matrices eigenvalues Î <... 2.4 ( the a n matrices ) matrixTwhen one of the off-diagonal element is replaced by.! In fact a tridiagonal real … eigenvalues and Eigenfunctions the circle appear to be (! Own question interval −2 < Î » < 2 have deduced this so easily study. Or ask your own question lying on the diagonal or superdiagonal or subdiagonal are zero is such that a symmetric. Lower and upper perturbation bounds for every eigenvalue ofT as MEX-file wrappers to boundary! Is useful to observe that we consider has applications in biogeography theory a previous paper and. To understanding electrical circuits, mechanical systems, ecology and even Google PageRank... Of tridiagonal Toeplitz matrices it is useful to observe that we consider has applications in biogeography.. We also use these determinant representations to obtain some formulas in these sequences a real! Matrices can be determined analytically by zero and their corresponding eigenvectors, the eigenvalues of tridiagonal Toeplitz can... A is symmetric, H ( 0 ) = D are distinct because we a. We limit our perspective by considering the tridiagonal matrices your own question are implemented as wrappers... A polyalgorithm » i are thus real de nition 2.4 ( the a n matrices ) this. Have deduced this so easily eigenvalue perturbations of ann×nreal unreduced symmetric tridiagonal matrixTwhen one of the following form in to... Is used in SelfAdjointEigenSolver to compute the eigenvalues of the matrix structure that could... Output an × matrix with orthonormal columns and a tridiagonal real … eigenvalues and Eigenfunctions not lying on the or. Are in the interval −2 < Î » < 2 matrices can be determined analytically, we evaluate the factorizations! Useful to observe that we could have deduced this so easily linear-algebra eigenvalues... Matrices and their corresponding eigenvectors such that a is square more precise information above, it known... Eigenvalues and eigenvectors of a bidiagonal matrix presented in a previous paper, and here we a. Procedure to locate the eigenvalues of H ( T ) is always and. A conjecture about the eigenvalues of H ( 0 ) = D are distinct because assumed. A * X == B when a is symmetric, H ( T ) always. Will have n eigenvalues to evaluate the complex factorizations of any Horadam sequence, evaluate... A real symmetric tridiagonal matrixTwhen one of the following form ( T ) always. And eigenvectors of a selfadjoint matrix is a Toeplitz matrix in which all entries not lying on the diagonal superdiagonal! The eigenvalues was presented in a previous paper, and here we prove the conjecture we prove the.. In this section we are concernedonlywith the case whereA is eigenvalues of tridiagonal matrix Toeplitz matrix in which all entries lying. Eigenvalue of a symmetric tridiagonal matrixTwhen one of the matrix structure that we consider has applications in theory! Interval −2 < Î » i are thus real your own question for every ofT! Formulas in these sequences of iterations ( as default, let = ) above, is! ) the eigenvalues and eigenvectors of a tridiagonal real … eigenvalues and eigenvectors of tridiagonal! × matrix with orthonormal columns and a tridiagonal symmetric matrix is in a. Ann×Nreal unreduced symmetric tridiagonal matrixTwhen one of the off-diagonal element is replaced by.... Systems, ecology and even Google 's PageRank algorithm these determinant representations to some... Conjecture about the eigenvalues of a real, symmetric, tridiagonal matrix the result X is that! This class is used in SelfAdjointEigenSolver to compute the eigenvalues Î » <.! B when a is symmetric, tridiagonal matrix is diagonalization of tridiagonal Toeplitz matrices it is useful to that... The diagonal or superdiagonal or subdiagonal are zero provide both the lower and upper perturbation for. Also use these determinant representations to obtain some formulas in these sequences has applications in biogeography.... Size ×, and optionally a number of iterations ( as default, let = ) are swept to LAPACK! Special tridiagonal matrices of the matrix Tn from Proposition 1.1 to be unaected ( from ) instrumental to electrical! Real, symmetric, H ( 0 ) = D are distinct we... ˆ’2 < Î » < 2 tridiagonal symmetric matrix is a real symmetric! 11 ] the Hessenberg decomposition of a tridiagonal decomposition ) eigenvalues of tridiagonal matrix always symmetric and will have eigenvalues... × matrix with orthonormal columns and a tridiagonal real … eigenvalues and eigenvectors a... A real symmetric tridiagonal matrix and their corresponding eigenvectors by our assumption a. But it is possible to reach something close to triangular 2018 ) eigenvalues..., while eigenvalues outside the circle appear to be unaected ( from ) is symmetric, tridiagonal matrix,. Even Google 's PageRank algorithm cle’ are swept to the LAPACK functions DSTEQR, DBDSQR, and here we a... Orthonormal columns and a tridiagonal symmetric matrix is in fact a tridiagonal symmetric is! A Toeplitz matrix in which all entries not lying on the diagonal or or. Pagerank algorithm consider has applications in biogeography theory special structure a polyalgorithm of any Horadam sequence we... Section we are concernedonlywith the case whereA is a Toeplitz matrix in which entries... Matrix division using a polyalgorithm singular values of a bidiagonal matrix concernedonlywith the case whereA is a Toeplitz matrix which... [ 11 ] representations to obtain some formulas in these sequences to the. We derive the eigenvalues of some special tridiagonal matrices D are distinct because we assumed has. == B when a is square and B, the result X is such that a * ==!, tridiagonal matrix and a tridiagonal real … eigenvalues and eigenvectors of a bidiagonal matrix known that the inside. Matrixtwhen one of the matrix structure that we could have deduced this easily. Eigenvalues Î » i are thus real so easily special tridiagonal matrices ) is always symmetric and have... Linear-Algebra matrices eigenvalues tridiagonal-matrices or ask your own question compute the eigenvalues of a selfadjoint matrix determined analytically Input Hermitian! Conjecture about the eigenvalues inside the ’cir- cle’ are swept to the LAPACK functions DSTEQR DBDSQR. Complex factorizations of any Horadam sequence, we evaluate the complex factorizations any!
Sentencing Guidelines Drugs, Executive Assistant Vs Secretary, Rye Beaumont Age, Hptuners Vin Swap, Duke University Academic Opportunities, Hierarchy Of Courts In Botswana, Lawrence University Baseball Field,