USAGE: rcond, info = NumRu::Lapack.cptcon( d, e, anorm, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CPTCON( N, D, E, ANORM, RCOND, RWORK, INFO ) * Purpose * ======= * * CPTCON computes the reciprocal of the condition number (in the * 1-norm) of a complex Hermitian positive definite tridiagonal matrix * using the factorization A = L*D*L**H or A = U**H*D*U computed by * CPTTRF. * * Norm(inv(A)) is computed by a direct method, and the reciprocal of * the condition number is computed as * RCOND = 1 / (ANORM * norm(inv(A))). * * Arguments * ========= * * N (input) INTEGER * The order of the matrix A. N >= 0. * * D (input) REAL array, dimension (N) * The n diagonal elements of the diagonal matrix D from the * factorization of A, as computed by CPTTRF. * * E (input) COMPLEX array, dimension (N-1) * The (n-1) off-diagonal elements of the unit bidiagonal factor * U or L from the factorization of A, as computed by CPTTRF. * * ANORM (input) REAL * The 1-norm of the original matrix A. * * RCOND (output) REAL * The reciprocal of the condition number of the matrix A, * computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the * 1-norm of inv(A) computed in this routine. * * RWORK (workspace) REAL array, dimension (N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * Further Details * =============== * * The method used is described in Nicholas J. Higham, "Efficient * Algorithms for Computing the Condition Number of a Tridiagonal * Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986. * * ===================================================================== *go to the page top

USAGE: info, d, e, z = NumRu::Lapack.cpteqr( compz, d, e, z, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) * Purpose * ======= * * CPTEQR computes all eigenvalues and, optionally, eigenvectors of a * symmetric positive definite tridiagonal matrix by first factoring the * matrix using SPTTRF and then calling CBDSQR to compute the singular * values of the bidiagonal factor. * * This routine computes the eigenvalues of the positive definite * tridiagonal matrix to high relative accuracy. This means that if the * eigenvalues range over many orders of magnitude in size, then the * small eigenvalues and corresponding eigenvectors will be computed * more accurately than, for example, with the standard QR method. * * The eigenvectors of a full or band positive definite Hermitian matrix * can also be found if CHETRD, CHPTRD, or CHBTRD has been used to * reduce this matrix to tridiagonal form. (The reduction to * tridiagonal form, however, may preclude the possibility of obtaining * high relative accuracy in the small eigenvalues of the original * matrix, if these eigenvalues range over many orders of magnitude.) * * Arguments * ========= * * COMPZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only. * = 'V': Compute eigenvectors of original Hermitian * matrix also. Array Z contains the unitary matrix * used to reduce the original matrix to tridiagonal * form. * = 'I': Compute eigenvectors of tridiagonal matrix also. * * N (input) INTEGER * The order of the matrix. N >= 0. * * D (input/output) REAL array, dimension (N) * On entry, the n diagonal elements of the tridiagonal matrix. * On normal exit, D contains the eigenvalues, in descending * order. * * E (input/output) REAL array, dimension (N-1) * On entry, the (n-1) subdiagonal elements of the tridiagonal * matrix. * On exit, E has been destroyed. * * Z (input/output) COMPLEX array, dimension (LDZ, N) * On entry, if COMPZ = 'V', the unitary matrix used in the * reduction to tridiagonal form. * On exit, if COMPZ = 'V', the orthonormal eigenvectors of the * original Hermitian matrix; * if COMPZ = 'I', the orthonormal eigenvectors of the * tridiagonal matrix. * If INFO > 0 on exit, Z contains the eigenvectors associated * with only the stored eigenvalues. * If COMPZ = 'N', then Z is not referenced. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * COMPZ = 'V' or 'I', LDZ >= max(1,N). * * WORK (workspace) REAL array, dimension (4*N) * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * > 0: if INFO = i, and i is: * <= N the Cholesky factorization of the matrix could * not be performed because the i-th principal minor * was not positive definite. * > N the SVD algorithm failed to converge; * if INFO = N+i, i off-diagonal elements of the * bidiagonal factor did not converge to zero. * * ==================================================================== *go to the page top

USAGE: ferr, berr, info, x = NumRu::Lapack.cptrfs( uplo, d, e, df, ef, b, x, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CPTRFS( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO ) * Purpose * ======= * * CPTRFS improves the computed solution to a system of linear * equations when the coefficient matrix is Hermitian positive definite * and tridiagonal, and provides error bounds and backward error * estimates for the solution. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the superdiagonal or the subdiagonal of the * tridiagonal matrix A is stored and the form of the * factorization: * = 'U': E is the superdiagonal of A, and A = U**H*D*U; * = 'L': E is the subdiagonal of A, and A = L*D*L**H. * (The two forms are equivalent if A is real.) * * N (input) INTEGER * The order of the matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrix B. NRHS >= 0. * * D (input) REAL array, dimension (N) * The n real diagonal elements of the tridiagonal matrix A. * * E (input) COMPLEX array, dimension (N-1) * The (n-1) off-diagonal elements of the tridiagonal matrix A * (see UPLO). * * DF (input) REAL array, dimension (N) * The n diagonal elements of the diagonal matrix D from * the factorization computed by CPTTRF. * * EF (input) COMPLEX array, dimension (N-1) * The (n-1) off-diagonal elements of the unit bidiagonal * factor U or L from the factorization computed by CPTTRF * (see UPLO). * * B (input) COMPLEX array, dimension (LDB,NRHS) * The right hand side matrix B. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * X (input/output) COMPLEX array, dimension (LDX,NRHS) * On entry, the solution matrix X, as computed by CPTTRS. * On exit, the improved solution matrix X. * * LDX (input) INTEGER * The leading dimension of the array X. LDX >= max(1,N). * * FERR (output) REAL array, dimension (NRHS) * The forward error bound for each solution vector * X(j) (the j-th column of the solution matrix X). * If XTRUE is the true solution corresponding to X(j), FERR(j) * is an estimated upper bound for the magnitude of the largest * element in (X(j) - XTRUE) divided by the magnitude of the * largest element in X(j). * * BERR (output) REAL array, dimension (NRHS) * The componentwise relative backward error of each solution * vector X(j) (i.e., the smallest relative change in * any element of A or B that makes X(j) an exact solution). * * WORK (workspace) COMPLEX array, dimension (N) * * RWORK (workspace) REAL array, dimension (N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * Internal Parameters * =================== * * ITMAX is the maximum number of steps of iterative refinement. * * ===================================================================== *go to the page top

USAGE: info, d, e, b = NumRu::Lapack.cptsv( d, e, b, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CPTSV( N, NRHS, D, E, B, LDB, INFO ) * Purpose * ======= * * CPTSV computes the solution to a complex system of linear equations * A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal * matrix, and X and B are N-by-NRHS matrices. * * A is factored as A = L*D*L**H, and the factored form of A is then * used to solve the system of equations. * * Arguments * ========= * * N (input) INTEGER * The order of the matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrix B. NRHS >= 0. * * D (input/output) REAL array, dimension (N) * On entry, the n diagonal elements of the tridiagonal matrix * A. On exit, the n diagonal elements of the diagonal matrix * D from the factorization A = L*D*L**H. * * E (input/output) COMPLEX array, dimension (N-1) * On entry, the (n-1) subdiagonal elements of the tridiagonal * matrix A. On exit, the (n-1) subdiagonal elements of the * unit bidiagonal factor L from the L*D*L**H factorization of * A. E can also be regarded as the superdiagonal of the unit * bidiagonal factor U from the U**H*D*U factorization of A. * * B (input/output) COMPLEX array, dimension (LDB,NRHS) * On entry, the N-by-NRHS right hand side matrix B. * On exit, if INFO = 0, the N-by-NRHS solution matrix X. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the leading minor of order i is not * positive definite, and the solution has not been * computed. The factorization has not been completed * unless i = N. * * ===================================================================== * * .. External Subroutines .. EXTERNAL CPTTRF, CPTTRS, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX * ..go to the page top

USAGE: x, rcond, ferr, berr, info, df, ef = NumRu::Lapack.cptsvx( fact, d, e, df, ef, b, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO ) * Purpose * ======= * * CPTSVX uses the factorization A = L*D*L**H to compute the solution * to a complex system of linear equations A*X = B, where A is an * N-by-N Hermitian positive definite tridiagonal matrix and X and B * are N-by-NRHS matrices. * * Error bounds on the solution and a condition estimate are also * provided. * * Description * =========== * * The following steps are performed: * * 1. If FACT = 'N', the matrix A is factored as A = L*D*L**H, where L * is a unit lower bidiagonal matrix and D is diagonal. The * factorization can also be regarded as having the form * A = U**H*D*U. * * 2. If the leading i-by-i principal minor is not positive definite, * then the routine returns with INFO = i. Otherwise, the factored * form of A is used to estimate the condition number of the matrix * A. If the reciprocal of the condition number is less than machine * precision, INFO = N+1 is returned as a warning, but the routine * still goes on to solve for X and compute error bounds as * described below. * * 3. The system of equations is solved for X using the factored form * of A. * * 4. Iterative refinement is applied to improve the computed solution * matrix and calculate error bounds and backward error estimates * for it. * * Arguments * ========= * * FACT (input) CHARACTER*1 * Specifies whether or not the factored form of the matrix * A is supplied on entry. * = 'F': On entry, DF and EF contain the factored form of A. * D, E, DF, and EF will not be modified. * = 'N': The matrix A will be copied to DF and EF and * factored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrices B and X. NRHS >= 0. * * D (input) REAL array, dimension (N) * The n diagonal elements of the tridiagonal matrix A. * * E (input) COMPLEX array, dimension (N-1) * The (n-1) subdiagonal elements of the tridiagonal matrix A. * * DF (input or output) REAL array, dimension (N) * If FACT = 'F', then DF is an input argument and on entry * contains the n diagonal elements of the diagonal matrix D * from the L*D*L**H factorization of A. * If FACT = 'N', then DF is an output argument and on exit * contains the n diagonal elements of the diagonal matrix D * from the L*D*L**H factorization of A. * * EF (input or output) COMPLEX array, dimension (N-1) * If FACT = 'F', then EF is an input argument and on entry * contains the (n-1) subdiagonal elements of the unit * bidiagonal factor L from the L*D*L**H factorization of A. * If FACT = 'N', then EF is an output argument and on exit * contains the (n-1) subdiagonal elements of the unit * bidiagonal factor L from the L*D*L**H factorization of A. * * B (input) COMPLEX array, dimension (LDB,NRHS) * The N-by-NRHS right hand side matrix B. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * X (output) COMPLEX array, dimension (LDX,NRHS) * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. * * LDX (input) INTEGER * The leading dimension of the array X. LDX >= max(1,N). * * RCOND (output) REAL * The reciprocal condition number of the matrix A. If RCOND * is less than the machine precision (in particular, if * RCOND = 0), the matrix is singular to working precision. * This condition is indicated by a return code of INFO > 0. * * FERR (output) REAL array, dimension (NRHS) * The forward error bound for each solution vector * X(j) (the j-th column of the solution matrix X). * If XTRUE is the true solution corresponding to X(j), FERR(j) * is an estimated upper bound for the magnitude of the largest * element in (X(j) - XTRUE) divided by the magnitude of the * largest element in X(j). * * BERR (output) REAL array, dimension (NRHS) * The componentwise relative backward error of each solution * vector X(j) (i.e., the smallest relative change in any * element of A or B that makes X(j) an exact solution). * * WORK (workspace) COMPLEX array, dimension (N) * * RWORK (workspace) REAL array, dimension (N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, and i is * <= N: the leading minor of order i of A is * not positive definite, so the factorization * could not be completed, and the solution has not * been computed. RCOND = 0 is returned. * = N+1: U is nonsingular, but RCOND is less than machine * precision, meaning that the matrix is singular * to working precision. Nevertheless, the * solution and error bounds are computed because * there are a number of situations where the * computed solution can be more accurate than the * value of RCOND would suggest. * * ===================================================================== *go to the page top

USAGE: info, d, e = NumRu::Lapack.cpttrf( d, e, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CPTTRF( N, D, E, INFO ) * Purpose * ======= * * CPTTRF computes the L*D*L' factorization of a complex Hermitian * positive definite tridiagonal matrix A. The factorization may also * be regarded as having the form A = U'*D*U. * * Arguments * ========= * * N (input) INTEGER * The order of the matrix A. N >= 0. * * D (input/output) REAL array, dimension (N) * On entry, the n diagonal elements of the tridiagonal matrix * A. On exit, the n diagonal elements of the diagonal matrix * D from the L*D*L' factorization of A. * * E (input/output) COMPLEX array, dimension (N-1) * On entry, the (n-1) subdiagonal elements of the tridiagonal * matrix A. On exit, the (n-1) subdiagonal elements of the * unit bidiagonal factor L from the L*D*L' factorization of A. * E can also be regarded as the superdiagonal of the unit * bidiagonal factor U from the U'*D*U factorization of A. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -k, the k-th argument had an illegal value * > 0: if INFO = k, the leading minor of order k is not * positive definite; if k < N, the factorization could not * be completed, while if k = N, the factorization was * completed, but D(N) <= 0. * * ===================================================================== *go to the page top

USAGE: info, b = NumRu::Lapack.cpttrs( uplo, d, e, b, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CPTTRS( UPLO, N, NRHS, D, E, B, LDB, INFO ) * Purpose * ======= * * CPTTRS solves a tridiagonal system of the form * A * X = B * using the factorization A = U'*D*U or A = L*D*L' computed by CPTTRF. * D is a diagonal matrix specified in the vector D, U (or L) is a unit * bidiagonal matrix whose superdiagonal (subdiagonal) is specified in * the vector E, and X and B are N by NRHS matrices. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies the form of the factorization and whether the * vector E is the superdiagonal of the upper bidiagonal factor * U or the subdiagonal of the lower bidiagonal factor L. * = 'U': A = U'*D*U, E is the superdiagonal of U * = 'L': A = L*D*L', E is the subdiagonal of L * * N (input) INTEGER * The order of the tridiagonal matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrix B. NRHS >= 0. * * D (input) REAL array, dimension (N) * The n diagonal elements of the diagonal matrix D from the * factorization A = U'*D*U or A = L*D*L'. * * E (input) COMPLEX array, dimension (N-1) * If UPLO = 'U', the (n-1) superdiagonal elements of the unit * bidiagonal factor U from the factorization A = U'*D*U. * If UPLO = 'L', the (n-1) subdiagonal elements of the unit * bidiagonal factor L from the factorization A = L*D*L'. * * B (input/output) REAL array, dimension (LDB,NRHS) * On entry, the right hand side vectors B for the system of * linear equations. * On exit, the solution vectors, X. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -k, the k-th argument had an illegal value * * ===================================================================== * * .. Local Scalars .. LOGICAL UPPER INTEGER IUPLO, J, JB, NB * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. External Subroutines .. EXTERNAL CPTTS2, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * ..go to the page top

USAGE: b = NumRu::Lapack.cptts2( iuplo, d, e, b, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CPTTS2( IUPLO, N, NRHS, D, E, B, LDB ) * Purpose * ======= * * CPTTS2 solves a tridiagonal system of the form * A * X = B * using the factorization A = U'*D*U or A = L*D*L' computed by CPTTRF. * D is a diagonal matrix specified in the vector D, U (or L) is a unit * bidiagonal matrix whose superdiagonal (subdiagonal) is specified in * the vector E, and X and B are N by NRHS matrices. * * Arguments * ========= * * IUPLO (input) INTEGER * Specifies the form of the factorization and whether the * vector E is the superdiagonal of the upper bidiagonal factor * U or the subdiagonal of the lower bidiagonal factor L. * = 1: A = U'*D*U, E is the superdiagonal of U * = 0: A = L*D*L', E is the subdiagonal of L * * N (input) INTEGER * The order of the tridiagonal matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrix B. NRHS >= 0. * * D (input) REAL array, dimension (N) * The n diagonal elements of the diagonal matrix D from the * factorization A = U'*D*U or A = L*D*L'. * * E (input) COMPLEX array, dimension (N-1) * If IUPLO = 1, the (n-1) superdiagonal elements of the unit * bidiagonal factor U from the factorization A = U'*D*U. * If IUPLO = 0, the (n-1) subdiagonal elements of the unit * bidiagonal factor L from the factorization A = L*D*L'. * * B (input/output) REAL array, dimension (LDB,NRHS) * On entry, the right hand side vectors B for the system of * linear equations. * On exit, the solution vectors, X. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * ===================================================================== * * .. Local Scalars .. INTEGER I, J * .. * .. External Subroutines .. EXTERNAL CSSCAL * .. * .. Intrinsic Functions .. INTRINSIC CONJG * ..go to the page top

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