f03bff computes the determinant of a real symmetric positive definite matrix . f07fdf must be called first to supply the symmetric matrix in Cholesky factorized form. The storage (upper or lower triangular) used by f07fdf is not relevant to f03bff since only the diagonal elements of the factorized are referenced.
The routine may be called by the names f03bff or nagf_det_real_sym.
f03bff computes the determinant of a real symmetric positive definite matrix that has been factorized as , where is upper triangular, or , where is lower triangular. The determinant is the product of the squares of the diagonal elements of or . The Cholesky factorized form of the matrix must be supplied; this is returned by a call to f07fdf.
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag
1: – IntegerInput
On entry: , the order of the matrix .
2: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a
must be at least
On entry: the lower or upper triangle of the Cholesky factorized form of the positive definite symmetric matrix . Only the diagonal elements are referenced.
3: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f03bff is called.
4: – Real (Kind=nag_wp)Output
5: – IntegerOutput
On exit: the determinant of is given by . It is given in this form to avoid overflow or underflow.
6: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
On entry, and .
The matrix is not positive definite.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
The accuracy of the determinant depends on the conditioning of the original matrix. For a detailed error analysis see page 25 of Wilkinson and Reinsch (1971).
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f03bff is not threaded in any implementation.
The time taken by f03bff is approximately proportional to .
This example computes a Cholesky factorization and calculates the determinant of the real symmetric positive definite matrix