COMPLEX*16 or DOUBLE COMPLEX routines for symmetric or Hermitian positive definite band matrix
zpbcon
USAGE:
rcond, info = NumRu::Lapack.zpbcon( uplo, kd, ab, anorm, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE ZPBCON( UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, RWORK, INFO )
* Purpose
* =======
*
* ZPBCON estimates the reciprocal of the condition number (in the
* 1-norm) of a complex Hermitian positive definite band matrix using
* the Cholesky factorization A = U**H*U or A = L*L**H computed by
* ZPBTRF.
*
* An estimate is obtained for norm(inv(A)), and the reciprocal of the
* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangular factor stored in AB;
* = 'L': Lower triangular factor stored in AB.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
*
* AB (input) COMPLEX*16 array, dimension (LDAB,N)
* The triangular factor U or L from the Cholesky factorization
* A = U**H*U or A = L*L**H of the band matrix A, stored in the
* first KD+1 rows of the array. The j-th column of U or L is
* stored in the j-th column of the array AB as follows:
* if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
* if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* ANORM (input) DOUBLE PRECISION
* The 1-norm (or infinity-norm) of the Hermitian band matrix A.
*
* RCOND (output) DOUBLE PRECISION
* The reciprocal of the condition number of the matrix A,
* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
* estimate of the 1-norm of inv(A) computed in this routine.
*
* WORK (workspace) COMPLEX*16 array, dimension (2*N)
*
* RWORK (workspace) DOUBLE PRECISION array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
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zpbequ
USAGE:
s, scond, amax, info = NumRu::Lapack.zpbequ( uplo, kd, ab, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
* Purpose
* =======
*
* ZPBEQU computes row and column scalings intended to equilibrate a
* Hermitian positive definite band matrix A and reduce its condition
* number (with respect to the two-norm). S contains the scale factors,
* S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
* elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
* choice of S puts the condition number of B within a factor N of the
* smallest possible condition number over all possible diagonal
* scalings.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangular of A is stored;
* = 'L': Lower triangular of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* AB (input) COMPLEX*16 array, dimension (LDAB,N)
* The upper or lower triangle of the Hermitian band matrix A,
* stored in the first KD+1 rows of the array. The j-th column
* of A is stored in the j-th column of the array AB as follows:
* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*
* LDAB (input) INTEGER
* The leading dimension of the array A. LDAB >= KD+1.
*
* S (output) DOUBLE PRECISION array, dimension (N)
* If INFO = 0, S contains the scale factors for A.
*
* SCOND (output) DOUBLE PRECISION
* If INFO = 0, S contains the ratio of the smallest S(i) to
* the largest S(i). If SCOND >= 0.1 and AMAX is neither too
* large nor too small, it is not worth scaling by S.
*
* AMAX (output) DOUBLE PRECISION
* Absolute value of largest matrix element. If AMAX is very
* close to overflow or very close to underflow, the matrix
* should be scaled.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = i, the i-th diagonal element is nonpositive.
*
* =====================================================================
*
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zpbrfs
USAGE:
ferr, berr, info, x = NumRu::Lapack.zpbrfs( uplo, kd, ab, afb, b, x, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
* Purpose
* =======
*
* ZPBRFS improves the computed solution to a system of linear
* equations when the coefficient matrix is Hermitian positive definite
* and banded, and provides error bounds and backward error estimates
* for the solution.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
* The upper or lower triangle of the Hermitian band matrix A,
* stored in the first KD+1 rows of the array. The j-th column
* of A is stored in the j-th column of the array AB as follows:
* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* AFB (input) COMPLEX*16 array, dimension (LDAFB,N)
* The triangular factor U or L from the Cholesky factorization
* A = U**H*U or A = L*L**H of the band matrix A as computed by
* ZPBTRF, in the same storage format as A (see AB).
*
* LDAFB (input) INTEGER
* The leading dimension of the array AFB. LDAFB >= KD+1.
*
* B (input) COMPLEX*16 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*16 array, dimension (LDX,NRHS)
* On entry, the solution matrix X, as computed by ZPBTRS.
* On exit, the improved solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* FERR (output) DOUBLE PRECISION array, dimension (NRHS)
* The estimated 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). The estimate is as reliable as
* the estimate for RCOND, and is almost always a slight
* overestimate of the true error.
*
* BERR (output) DOUBLE PRECISION 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*16 array, dimension (2*N)
*
* RWORK (workspace) DOUBLE PRECISION 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.
*
* =====================================================================
*
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zpbstf
USAGE:
info, ab = NumRu::Lapack.zpbstf( uplo, kd, ab, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )
* Purpose
* =======
*
* ZPBSTF computes a split Cholesky factorization of a complex
* Hermitian positive definite band matrix A.
*
* This routine is designed to be used in conjunction with ZHBGST.
*
* The factorization has the form A = S**H*S where S is a band matrix
* of the same bandwidth as A and the following structure:
*
* S = ( U )
* ( M L )
*
* where U is upper triangular of order m = (n+kd)/2, and L is lower
* triangular of order n-m.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
* On entry, the upper or lower triangle of the Hermitian band
* matrix A, stored in the first kd+1 rows of the array. The
* j-th column of A is stored in the j-th column of the array AB
* as follows:
* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*
* On exit, if INFO = 0, the factor S from the split Cholesky
* factorization A = S**H*S. See Further Details.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, the factorization could not be completed,
* because the updated element a(i,i) was negative; the
* matrix A is not positive definite.
*
* Further Details
* ===============
*
* The band storage scheme is illustrated by the following example, when
* N = 7, KD = 2:
*
* S = ( s11 s12 s13 )
* ( s22 s23 s24 )
* ( s33 s34 )
* ( s44 )
* ( s53 s54 s55 )
* ( s64 s65 s66 )
* ( s75 s76 s77 )
*
* If UPLO = 'U', the array AB holds:
*
* on entry: on exit:
*
* * * a13 a24 a35 a46 a57 * * s13 s24 s53' s64' s75'
* * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54' s65' s76'
* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
*
* If UPLO = 'L', the array AB holds:
*
* on entry: on exit:
*
* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
* a21 a32 a43 a54 a65 a76 * s12' s23' s34' s54 s65 s76 *
* a31 a42 a53 a64 a64 * * s13' s24' s53 s64 s75 * *
*
* Array elements marked * are not used by the routine; s12' denotes
* conjg(s12); the diagonal elements of S are real.
*
* =====================================================================
*
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zpbsv
USAGE:
info, ab, b = NumRu::Lapack.zpbsv( uplo, kd, ab, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE ZPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
* Purpose
* =======
*
* ZPBSV computes the solution to a complex system of linear equations
* A * X = B,
* where A is an N-by-N Hermitian positive definite band matrix and X
* and B are N-by-NRHS matrices.
*
* The Cholesky decomposition is used to factor A as
* A = U**H * U, if UPLO = 'U', or
* A = L * L**H, if UPLO = 'L',
* where U is an upper triangular band matrix, and L is a lower
* triangular band matrix, with the same number of superdiagonals or
* subdiagonals as A. The factored form of A is then used to solve the
* system of equations A * X = B.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The number of linear equations, i.e., the order of the
* matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
* On entry, the upper or lower triangle of the Hermitian band
* matrix A, stored in the first KD+1 rows of the array. The
* j-th column of A is stored in the j-th column of the array AB
* as follows:
* if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
* See below for further details.
*
* On exit, if INFO = 0, the triangular factor U or L from the
* Cholesky factorization A = U**H*U or A = L*L**H of the band
* matrix A, in the same storage format as A.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* B (input/output) COMPLEX*16 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 of A is not
* positive definite, so the factorization could not be
* completed, and the solution has not been computed.
*
* Further Details
* ===============
*
* The band storage scheme is illustrated by the following example, when
* N = 6, KD = 2, and UPLO = 'U':
*
* On entry: On exit:
*
* * * a13 a24 a35 a46 * * u13 u24 u35 u46
* * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
* a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*
* Similarly, if UPLO = 'L' the format of A is as follows:
*
* On entry: On exit:
*
* a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
* a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
* a31 a42 a53 a64 * * l31 l42 l53 l64 * *
*
* Array elements marked * are not used by the routine.
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZPBTRF, ZPBTRS
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
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zpbsvx
USAGE:
x, rcond, ferr, berr, info, ab, afb, equed, s, b = NumRu::Lapack.zpbsvx( fact, uplo, kd, ab, afb, equed, s, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE ZPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
* Purpose
* =======
*
* ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*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 band 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 = 'E', real scaling factors are computed to equilibrate
* the system:
* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
* Whether or not the system will be equilibrated depends on the
* scaling of the matrix A, but if equilibration is used, A is
* overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*
* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
* factor the matrix A (after equilibration if FACT = 'E') as
* A = U**H * U, if UPLO = 'U', or
* A = L * L**H, if UPLO = 'L',
* where U is an upper triangular band matrix, and L is a lower
* triangular band matrix.
*
* 3. 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.
*
* 4. The system of equations is solved for X using the factored form
* of A.
*
* 5. Iterative refinement is applied to improve the computed solution
* matrix and calculate error bounds and backward error estimates
* for it.
*
* 6. If equilibration was used, the matrix X is premultiplied by
* diag(S) so that it solves the original system before
* equilibration.
*
* Arguments
* =========
*
* FACT (input) CHARACTER*1
* Specifies whether or not the factored form of the matrix A is
* supplied on entry, and if not, whether the matrix A should be
* equilibrated before it is factored.
* = 'F': On entry, AFB contains the factored form of A.
* If EQUED = 'Y', the matrix A has been equilibrated
* with scaling factors given by S. AB and AFB will not
* be modified.
* = 'N': The matrix A will be copied to AFB and factored.
* = 'E': The matrix A will be equilibrated if necessary, then
* copied to AFB and factored.
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The number of linear equations, i.e., the order of the
* matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* NRHS (input) INTEGER
* The number of right-hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
* On entry, the upper or lower triangle of the Hermitian band
* matrix A, stored in the first KD+1 rows of the array, except
* if FACT = 'F' and EQUED = 'Y', then A must contain the
* equilibrated matrix diag(S)*A*diag(S). The j-th column of A
* is stored in the j-th column of the array AB as follows:
* if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
* See below for further details.
*
* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
* diag(S)*A*diag(S).
*
* LDAB (input) INTEGER
* The leading dimension of the array A. LDAB >= KD+1.
*
* AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N)
* If FACT = 'F', then AFB is an input argument and on entry
* contains the triangular factor U or L from the Cholesky
* factorization A = U**H*U or A = L*L**H of the band matrix
* A, in the same storage format as A (see AB). If EQUED = 'Y',
* then AFB is the factored form of the equilibrated matrix A.
*
* If FACT = 'N', then AFB is an output argument and on exit
* returns the triangular factor U or L from the Cholesky
* factorization A = U**H*U or A = L*L**H.
*
* If FACT = 'E', then AFB is an output argument and on exit
* returns the triangular factor U or L from the Cholesky
* factorization A = U**H*U or A = L*L**H of the equilibrated
* matrix A (see the description of A for the form of the
* equilibrated matrix).
*
* LDAFB (input) INTEGER
* The leading dimension of the array AFB. LDAFB >= KD+1.
*
* EQUED (input or output) CHARACTER*1
* Specifies the form of equilibration that was done.
* = 'N': No equilibration (always true if FACT = 'N').
* = 'Y': Equilibration was done, i.e., A has been replaced by
* diag(S) * A * diag(S).
* EQUED is an input argument if FACT = 'F'; otherwise, it is an
* output argument.
*
* S (input or output) DOUBLE PRECISION array, dimension (N)
* The scale factors for A; not accessed if EQUED = 'N'. S is
* an input argument if FACT = 'F'; otherwise, S is an output
* argument. If FACT = 'F' and EQUED = 'Y', each element of S
* must be positive.
*
* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS right hand side matrix B.
* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
* B is overwritten by diag(S) * B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (output) COMPLEX*16 array, dimension (LDX,NRHS)
* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
* the original system of equations. Note that if EQUED = 'Y',
* A and B are modified on exit, and the solution to the
* equilibrated system is inv(diag(S))*X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* RCOND (output) DOUBLE PRECISION
* The estimate of the reciprocal condition number of the matrix
* A after equilibration (if done). 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) DOUBLE PRECISION array, dimension (NRHS)
* The estimated 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). The estimate is as reliable as
* the estimate for RCOND, and is almost always a slight
* overestimate of the true error.
*
* BERR (output) DOUBLE PRECISION 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*16 array, dimension (2*N)
*
* RWORK (workspace) DOUBLE PRECISION 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.
*
* Further Details
* ===============
*
* The band storage scheme is illustrated by the following example, when
* N = 6, KD = 2, and UPLO = 'U':
*
* Two-dimensional storage of the Hermitian matrix A:
*
* a11 a12 a13
* a22 a23 a24
* a33 a34 a35
* a44 a45 a46
* a55 a56
* (aij=conjg(aji)) a66
*
* Band storage of the upper triangle of A:
*
* * * a13 a24 a35 a46
* * a12 a23 a34 a45 a56
* a11 a22 a33 a44 a55 a66
*
* Similarly, if UPLO = 'L' the format of A is as follows:
*
* a11 a22 a33 a44 a55 a66
* a21 a32 a43 a54 a65 *
* a31 a42 a53 a64 * *
*
* Array elements marked * are not used by the routine.
*
* =====================================================================
*
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zpbtf2
USAGE:
info, ab = NumRu::Lapack.zpbtf2( uplo, kd, ab, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE ZPBTF2( UPLO, N, KD, AB, LDAB, INFO )
* Purpose
* =======
*
* ZPBTF2 computes the Cholesky factorization of a complex Hermitian
* positive definite band matrix A.
*
* The factorization has the form
* A = U' * U , if UPLO = 'U', or
* A = L * L', if UPLO = 'L',
* where U is an upper triangular matrix, U' is the conjugate transpose
* of U, and L is lower triangular.
*
* This is the unblocked version of the algorithm, calling Level 2 BLAS.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the upper or lower triangular part of the
* Hermitian matrix A is stored:
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of super-diagonals of the matrix A if UPLO = 'U',
* or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
*
* AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
* On entry, the upper or lower triangle of the Hermitian band
* matrix A, stored in the first KD+1 rows of the array. The
* j-th column of A is stored in the j-th column of the array AB
* as follows:
* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*
* On exit, if INFO = 0, the triangular factor U or L from the
* Cholesky factorization A = U'*U or A = L*L' of the band
* matrix A, in the same storage format as A.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* 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, and the factorization could not be
* completed.
*
* Further Details
* ===============
*
* The band storage scheme is illustrated by the following example, when
* N = 6, KD = 2, and UPLO = 'U':
*
* On entry: On exit:
*
* * * a13 a24 a35 a46 * * u13 u24 u35 u46
* * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
* a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*
* Similarly, if UPLO = 'L' the format of A is as follows:
*
* On entry: On exit:
*
* a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
* a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
* a31 a42 a53 a64 * * l31 l42 l53 l64 * *
*
* Array elements marked * are not used by the routine.
*
* =====================================================================
*
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zpbtrf
USAGE:
info, ab = NumRu::Lapack.zpbtrf( uplo, kd, ab, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE ZPBTRF( UPLO, N, KD, AB, LDAB, INFO )
* Purpose
* =======
*
* ZPBTRF computes the Cholesky factorization of a complex Hermitian
* positive definite band matrix A.
*
* The factorization has the form
* A = U**H * U, if UPLO = 'U', or
* A = L * L**H, if UPLO = 'L',
* where U is an upper triangular matrix and L is lower triangular.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
* On entry, the upper or lower triangle of the Hermitian band
* matrix A, stored in the first KD+1 rows of the array. The
* j-th column of A is stored in the j-th column of the array AB
* as follows:
* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*
* On exit, if INFO = 0, the triangular factor U or L from the
* Cholesky factorization A = U**H*U or A = L*L**H of the band
* matrix A, in the same storage format as A.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* 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 factorization could not be
* completed.
*
* Further Details
* ===============
*
* The band storage scheme is illustrated by the following example, when
* N = 6, KD = 2, and UPLO = 'U':
*
* On entry: On exit:
*
* * * a13 a24 a35 a46 * * u13 u24 u35 u46
* * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
* a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*
* Similarly, if UPLO = 'L' the format of A is as follows:
*
* On entry: On exit:
*
* a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
* a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
* a31 a42 a53 a64 * * l31 l42 l53 l64 * *
*
* Array elements marked * are not used by the routine.
*
* Contributed by
* Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
*
* =====================================================================
*
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zpbtrs
USAGE:
info, b = NumRu::Lapack.zpbtrs( uplo, kd, ab, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE ZPBTRS( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
* Purpose
* =======
*
* ZPBTRS solves a system of linear equations A*X = B with a Hermitian
* positive definite band matrix A using the Cholesky factorization
* A = U**H*U or A = L*L**H computed by ZPBTRF.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangular factor stored in AB;
* = 'L': Lower triangular factor stored in AB.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* AB (input) COMPLEX*16 array, dimension (LDAB,N)
* The triangular factor U or L from the Cholesky factorization
* A = U**H*U or A = L*L**H of the band matrix A, stored in the
* first KD+1 rows of the array. The j-th column of U or L is
* stored in the j-th column of the array AB as follows:
* if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
* if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
* On entry, the right hand side matrix B.
* On exit, the 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
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZTBSV
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
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