f07ap uses the

L U

factorization to compute the solution to a complex system of linear equations

A X = B or A^{T} X = B or A^{H} X = B,

where

A

is an

n

n

matrix and

X

and

B

are

n

r

matrices. Error bounds on the solution and a condition estimate are also provided.

Syntax

C#
public static void f07ap( string fact, string trans, int n, int nrhs, Complex[,] a, Complex[,] af, int[] ipiv, ref string equed, double[] r, double[] c, Complex[,] b, Complex[,] x, out double rcond, double[] ferr, double[] berr, out double rgf, out int info )

public static void f07ap(
	string fact,
	string trans,
	int n,
	int nrhs,
	Complex[,] a,
	Complex[,] af,
	int[] ipiv,
	ref string equed,
	double[] r,
	double[] c,
	Complex[,] b,
	Complex[,] x,
	out double rcond,
	double[] ferr,
	double[] berr,
	out double rgf,
	out int info
)

Visual Basic
Public Shared Sub f07ap ( _ fact As String, _ trans As String, _ n As Integer, _ nrhs As Integer, _ a As Complex(,), _ af As Complex(,), _ ipiv As Integer(), _ ByRef equed As String, _ r As Double(), _ c As Double(), _ b As Complex(,), _ x As Complex(,), _ <OutAttribute> ByRef rcond As Double, _ ferr As Double(), _ berr As Double(), _ <OutAttribute> ByRef rgf As Double, _ <OutAttribute> ByRef info As Integer _ )

Visual Basic

Public Shared Sub f07ap ( _
	fact As String, _
	trans As String, _
	n As Integer, _
	nrhs As Integer, _
	a As Complex(,), _
	af As Complex(,), _
	ipiv As Integer(), _
	ByRef equed As String, _
	r As Double(), _
	c As Double(), _
	b As Complex(,), _
	x As Complex(,), _
	<OutAttribute> ByRef rcond As Double, _
	ferr As Double(), _
	berr As Double(), _
	<OutAttribute> ByRef rgf As Double, _
	<OutAttribute> ByRef info As Integer _
)

Visual C++
public: static void f07ap( String^ fact, String^ trans, int n, int nrhs, array<Complex,2>^ a, array<Complex,2>^ af, array<int>^ ipiv, String^% equed, array<double>^ r, array<double>^ c, array<Complex,2>^ b, array<Complex,2>^ x, [OutAttribute] double% rcond, array<double>^ ferr, array<double>^ berr, [OutAttribute] double% rgf, [OutAttribute] int% info )

Visual C++

public:
static void f07ap(
	String^ fact, 
	String^ trans, 
	int n, 
	int nrhs, 
	array<Complex,2>^ a, 
	array<Complex,2>^ af, 
	array<int>^ ipiv, 
	String^% equed, 
	array<double>^ r, 
	array<double>^ c, 
	array<Complex,2>^ b, 
	array<Complex,2>^ x, 
	[OutAttribute] double% rcond, 
	array<double>^ ferr, 
	array<double>^ berr, 
	[OutAttribute] double% rgf, 
	[OutAttribute] int% info
)

F#
static member f07ap : fact : string * trans : string * n : int * nrhs : int * a : Complex[,] * af : Complex[,] * ipiv : int[] * equed : string byref * r : float[] * c : float[] * b : Complex[,] * x : Complex[,] * rcond : float byref * ferr : float[] * berr : float[] * rgf : float byref * info : int byref -> unit

static member f07ap : 
        fact : string * 
        trans : string * 
        n : int * 
        nrhs : int * 
        a : Complex[,] * 
        af : Complex[,] * 
        ipiv : int[] * 
        equed : string byref * 
        r : float[] * 
        c : float[] * 
        b : Complex[,] * 
        x : Complex[,] * 
        rcond : float byref * 
        ferr : float[] * 
        berr : float[] * 
        rgf : float byref * 
        info : int byref -> unit

Parameters

fact

Type: System..::..String

On entry: specifies whether or not the factorized form of the matrix

A

is supplied on entry, and if not, whether the matrix

A

should be equilibrated before it is factorized.

$fact = "F"$: af and ipiv contain the factorized form of $A$ . If $equed \neq "N"$ , the matrix $A$ has been equilibrated with scaling factors given by r and c. a, af and ipiv are not modified.
$fact = "N"$: The matrix $A$ will be copied to af and factorized.
$fact = "E"$: The matrix $A$ will be equilibrated if necessary, then copied to af and factorized.

Constraint:

fact = "F"

"N"

"E"

trans

Type: System..::..String

On entry: specifies the form of the system of equations.

$trans = "N"$: $A X = B$ (No transpose).
$trans = "T"$: $A^{T} X = B$ (Transpose).
$trans = "C"$: $A^{H} X = B$ (Conjugate transpose).

Constraint:

trans = "N"

"T"

"C"

n: Type: System..::..Int32
On entry: $n$ , the number of linear equations, i.e., the order of the matrix $A$ .

Constraint: $n \geq 0$ .

nrhs: Type: System..::..Int32
On entry: $r$ , the number of right-hand sides, i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs \geq 0$ .

a

Type: array<NagLibrary..::..Complex,2>[,](,)[,][,]

An array of size [dim1, dim2]

Note: dim1 must satisfy the constraint:

dim1 \geq \max (1, n)

Note: the second dimension of the array a must be at least

\max (1, n)

On entry: the

n

n

matrix

A

fact = "F"

and

equed \neq "N"

, a must have been equilibrated by the scaling factors in r and/or c.

On exit: if

fact = "F"

"N"

, or if

fact = "E"

and

equed = "N"

, a is not modified.

fact = "E"

equed \neq "N"

A

is scaled as follows:

if $equed = "R"$ , $A = D_{R} A$ ;
if $equed = "C"$ , $A = A D_{C}$ ;
if $equed = "B"$ , $A = D_{R} A D_{C}$ .

af: Type: array<NagLibrary..::..Complex,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $dim1 \geq \max (1, n)$
Note: the second dimension of the array af must be at least $\max (1, n)$ .
On entry: if $fact = "F"$ , af contains the factors $L$ and $U$ from the factorization $A = P L U$ as computed by f07ar. If $equed \neq "N"$ , af is the factorized form of the equilibrated matrix $A$ .
If $fact = "N"$ or $"E"$ , af need not be set.

On exit: if $fact = "N"$ , af returns the factors $L$ and $U$ from the factorization $A = P L U$ of the original matrix $A$ .
If $fact = "E"$ , af returns the factors $L$ and $U$ from the factorization $A = P L U$ of the equilibrated matrix $A$ (see the description of a for the form of the equilibrated matrix).

If $fact = "F"$ , af is unchanged from entry.

ipiv: Type: array<System..::..Int32>[]()[][]
An array of size [dim1]
Note: the dimension of the array ipiv must be at least $\max (1, n)$ .
On entry: if $fact = "F"$ , ipiv contains the pivot indices from the factorization $A = P L U$ as computed by f07ar; at the $i$ th step row $i$ of the matrix was interchanged with row $ipiv [i - 1]$ . $ipiv [i - 1] = i$ indicates a row interchange was not required.
If $fact = "N"$ or $"E"$ , ipiv need not be set.

On exit: if $fact = "N"$ , ipiv contains the pivot indices from the factorization $A = P L U$ of the original matrix $A$ .
If $fact = "E"$ , ipiv contains the pivot indices from the factorization $A = P L U$ of the equilibrated matrix $A$ .

If $fact = "F"$ , ipiv is unchanged from entry.

equed

Type: System..::..String%

On entry: if

fact = "N"

"E"

, equed need not be set.

fact = "F"

, equed must specify the form of the equilibration that was performed as follows:

if $equed = "N"$ , no equilibration;
if $equed = "R"$ , row equilibration, i.e., $A$ has been premultiplied by $D_{R}$ ;
if $equed = "C"$ , column equilibration, i.e., $A$ has been postmultiplied by $D_{C}$ ;
if $equed = "B"$ , both row and column equilibration, i.e., $A$ has been replaced by $D_{R} A D_{C}$ .

On exit: if

fact = "F"

, equed is unchanged from entry.

Otherwise, if no constraints are violated, equed specifies the form of equilibration that was performed as specified above.

Constraint: if

fact = "F"

equed = "N"

"R"

"C"

"B"

r: Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension of the array r must be at least $\max (1, n)$ .
On entry: if $fact = "N"$ or $"E"$ , r need not be set.
If $fact = "F"$ and $equed = "R"$ or $"B"$ , r must contain the row scale factors for $A$ , $D_{R}$ ; each element of r must be positive.

On exit: if $fact = "F"$ , r is unchanged from entry.
Otherwise, if no constraints are violated and $equed = "R"$ or $"B"$ , r contains the row scale factors for $A$ , $D_{R}$ , such that $A$ is multiplied on the left by $D_{R}$ ; each element of r is positive.

c: Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension of the array c must be at least $\max (1, n)$ .
On entry: if $fact = "N"$ or $"E"$ , c need not be set.
If $fact = "F"$ or $equed = "C"$ or $"B"$ , c must contain the column scale factors for $A$ , $D_{C}$ ; each element of c must be positive.

On exit: if $fact = "F"$ , c is unchanged from entry.
Otherwise, if no constraints are violated and $equed = "C"$ or $"B"$ , c contains the row scale factors for $A$ , $D_{C}$ ; each element of c is positive.

b: Type: array<NagLibrary..::..Complex,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $dim1 \geq \max (1, n)$
Note: the second dimension of the array b must be at least $\max (1, nrhs)$ .
On entry: the $n$ by $r$ right-hand side matrix $B$ .
On exit: if $equed = "N"$ , b is not modified.
If $trans = "N"$ and $equed = "R"$ or $"B"$ , b is overwritten by $D_{R} B$ .

If $trans = "T"$ or $"C"$ and $equed = "C"$ or $"B"$ , b is overwritten by $D_{C} B$ .

x: Type: array<NagLibrary..::..Complex,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $dim1 \geq \max (1, n)$
Note: the second dimension of the array x must be at least $\max (1, nrhs)$ .
On exit: if $info = 0$ or $n + 1$ , the $n$ by $r$ solution matrix $X$ to the original system of equations. Note that the arrays $A$ and $B$ are modified on exit if $equed \neq "N"$ , and the solution to the equilibrated system is $D_{C}^{- 1} X$ if $trans = "N"$ and $equed = "C"$ or $"B"$ , or $D_{R}^{- 1} X$ if $trans = "T"$ or $"C"$ and $equed = "R"$ or $"B"$ .

rcond: Type: System..::..Double%
On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix $A$ (after equilibration if that is performed), computed as $rcond = 1.0 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})$ .

ferr: Type: array<System..::..Double>[]()[][]
An array of size [nrhs]
On exit: if $info = 0$ or $n + 1$ , an estimate of the forward error bound for each computed solution vector, such that ${‖{\hat{x}}_{j} - x_{j}‖}_{\infty} / {‖x_{j}‖}_{\infty} \leq ferr [j - 1]$ where ${\hat{x}}_{j}$ is the $j$ th column of the computed solution returned in the array x and $x_{j}$ is the corresponding column of the exact solution $X$ . The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.

berr: Type: array<System..::..Double>[]()[][]
An array of size [nrhs]
On exit: if $info = 0$ or $n + 1$ , an estimate of the component-wise relative backward error of each computed solution vector ${\hat{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_{j}$ an exact solution).

rgf: Type: System..::..Double%
On exit: if $info = 0$ , the reciprocal pivot growth factor $‖A‖ / ‖U‖$ , where $‖.‖$ denotes the maximum absolute element norm. If $rgf ≪ 1$ , the stability of the $L U$ factorization of $A$ could be poor. This also means that the solution x, condition estimate rcond, and forward error bound ferr could be unreliable. If the factorization fails with $info > 0 and info \leq n$ , then $rgf$ contains the reciprocal pivot growth factor for the leading info columns of $A$ .

info: Type: System..::..Int32%
On exit: $info = 0$ unless the method detects an error (see [Error Indicators and Warnings]).

Description

f07ap performs the following steps:

Equilibration

The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting

fact = "E"

. In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems

A X = B

A^{T} X = B

and

A^{H} X = B

are

(D_{R} A D_{C}) (D_{C}^{- 1} X) = D_{R} B,

{(D_{R} A D_{C})}^{T} (D_{R}^{- 1} X) = D_{C} B,

and

{(D_{R} A D_{C})}^{H} (D_{R}^{- 1} X) = D_{C} B,

respectively, where

D_{R}

and

D_{C}

are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.

When equilibration is used,

A

will be overwritten by

D_{R} A D_{C}

and

B

will be overwritten by

D_{R} B

(or

D_{C} B

when the solution of

A^{T} X = B

A^{H} X = B

is sought).

Factorization

The matrix

A

, or its scaled form, is copied and factored using the

L U

decomposition

A = P L U,

where

P

is a permutation matrix,

L

is a unit lower triangular matrix, and

U

is upper triangular.

This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to f07ap with the same matrix

A

Condition Number Estimation

The

L U

factorization of

A

determines whether a solution to the linear system exists. If some diagonal element of

U

is zero, then

U

is exactly singular, no solution exists and the method returns with a failure. Otherwise the factorized 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 then a warning code is returned on final exit.

Solution

The (equilibrated) system is solved for

X

(

D_{C}^{- 1} X

D_{R}^{- 1} X

) using the factored form of

A

(

D_{R} A D_{C}

Iterative Refinement

Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution.

Construct Solution Matrix $X$

If equilibration was used, the matrix

X

is premultiplied by

D_{C}

(if

trans = "N"

) or

D_{R}

(if

trans = "T"

"C"

) so that it solves the original system before equilibration.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Error Indicators and Warnings

Some error messages may refer to parameters that are dropped from this interface (LDA, LDAF, LDB, LDX, RWORK) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0 and info \leq n$: Element $〈value〉$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. $rcond = 0.0$ is returned.

$info = n + 1$: $U$ 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.

$ifail = -9000$: An error occured, see message report.
$ifail = -6000$: Invalid Parameters $〈value〉$
$ifail = -4000$: Invalid dimension for array $〈value〉$
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

For each right-hand side vector

b

, the computed solution

\hat{x}

is the exact solution of a perturbed system of equations

(A + E) \hat{x} = b

, where

|E| \leq c (n) ε P |L| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. See Section 9.3 of Higham (2002) for further details.

x

is the true solution, then the computed solution

\hat{x}

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖\hat{x}‖}_{\infty}} \leq w_{c} cond (A, \hat{x}, b)

where

cond (A, \hat{x}, b) = {‖|A^{- 1}| (|A| |\hat{x}| + |b|)‖}_{\infty} / {‖\hat{x}‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

. If

\hat{x}

is the

j

th column of

X

, then

w_{c}

is returned in

berr [j - 1]

and a bound on

{‖x - \hat{x}‖}_{\infty} / {‖\hat{x}‖}_{\infty}

is returned in

ferr [j - 1]

. See Section 4.4 of Anderson et al. (1999) for further details.

Parallelism and Performance

None.

Further Comments

The factorization of

A

requires approximately

\frac{8}{3} n^{3}

floating-point operations.

Estimating the forward error involves solving a number of systems of linear equations of the form

A x = b

A^{T} x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

8 n^{2}

operations.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The real analogue of this method is f07ab.

Example

This example solves the equations

A X = B,

where

A

is the general matrix

A = (\begin{array}{r} - 1.34 + 2.55 i & 0.28 + 3.17 i & - 6.39 - 2.20 i & 0.72 - 0.92 i \\ - 1.70 - 14.10 i & 33.10 - 1.50 i & - 1.50 + 13.40 i & 12.90 + 13.80 i \\ - 3.29 - 2.39 i & - 1.91 + 4.42 i & - 0.14 - 1.35 i & 1.72 + 1.35 i \\ 2.41 + 0.39 i & - 0.56 + 1.47 i & - 0.83 - 0.69 i & - 1.96 + 0.67 i \end{array})

and

B = (\begin{array}{r} 26.26 + 51.78 i & 31.32 - 6.70 i \\ 64.30 - 86.80 i & 158.60 - 14.20 i \\ - 5.75 + 25.31 i & - 2.15 + 30.19 i \\ 1.16 + 2.57 i & - 2.56 + 7.55 i \end{array}) .

Error estimates for the solutions, information on scaling, an estimate of the reciprocal of the condition number of the scaled matrix

A

and an estimate of the reciprocal of the pivot growth factor for the factorization of

A

are also output.

Example program (C#): f07ape.cs

Example program data: f07ape.d

Example program results: f07ape.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also