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NAG[f07hhc] NAG[nag_dpbrfs] - Refined solution with error bounds of real symmetric positive-definite band system of linear equations, multiple right-hand sides
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Calling Sequence
f07hhc(uplo, kd, nrhs, ab, afb, b, x, ferr, berr, 'n'=n, 'fail'=fail)
nag_dpbrfs(. . .)
Parameters
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uplo - String;
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On entry: indicates whether the upper or lower triangular part of is stored and how has been factorized.
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Constraint: "Nag_Upper" or "Nag_Lower". .
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kd - integer;
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On entry: , the number of superdiagonals or subdiagonals of the matrix .
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Constraint: . .
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nrhs - integer;
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On entry: , the number of right-hand sides.
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Constraint: . .
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ab - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);
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Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
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afb - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);
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Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
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b - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);
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Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
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x - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);
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Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
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On exit: the improved solution matrix .
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ferr - Vector(1..dim, datatype=float[8]);
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Note: the dimension, dim, of the array ferr must be at least .
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berr - Vector(1..dim, datatype=float[8]);
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Note: the dimension, dim, of the array berr must be at least .
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'n'=n - integer; (optional)
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Default value: the second dimension of the array ab.
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On entry: , the order of the matrix .
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Constraint: . .
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'fail'=fail - table; (optional)
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The NAG error argument, see the documentation for NagError.
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Description
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Purpose
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nag_dpbrfs (f07hhc) returns error bounds for the solution of a real symmetric positive-definite band system of linear equations with multiple right-hand sides, . It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.
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Error Indicators and Warnings
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"NE_ALLOC_FAIL"
Dynamic memory allocation failed.
"NE_BAD_PARAM"
On entry, argument had an illegal value.
"NE_INT"
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
"NE_INTERNAL_ERROR"
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.
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Accuracy
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The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.
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Further Comments
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For each right-hand side, computation of the backward error involves a minimum of floating-point operations. Each step of iterative refinement involves an additional operations. This assumes . At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form ; the number is usually 4 or 5 and never more than 11. Each solution involves approximately operations.
The complex analogue of this function is f07hvc (nag_zpbrfs).
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Examples
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>
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uplo := "Nag_Lower":
n := 4:
kd := 1:
nrhs := 2:
ab := Matrix([[0, 5.49, 2.68, 5.63], [-2.39, 2.6, -2.22, 5.17]], datatype=float[8], order='C_order'):
afb := Matrix([[0, 2.343074902771996, 1.143796127400537, 2.078877201506509], [-1.149659055507477, 1.130612248337004, -1.963537900164584, 1.146524711734229]], datatype=float[8], order='C_order'):
b := Matrix([[22.09, 5.1], [9.31, 30.81], [-5.24, -25.82], [11.83, 22.9]], datatype=float[8], order='C_order'):
x := Matrix([[5.000000000000002, -2.000000000000001], [-2.000000000000004, 6.000000000000002], [-3.000000000000006, -0.9999999999999974], [0.9999999999999974, 4.000000000000001]], datatype=float[8], order='C_order'):
ferr := Vector(2, datatype=float[8]):
berr := Vector(2, datatype=float[8]):
NAG:-f07hhc(uplo, kd, nrhs, ab, afb, b, x, ferr, berr, 'n' = n):
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