# zspsvx.f man page

zspsvx.f —

## Synopsis

### Functions/Subroutines

subroutinezspsvx(FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)ZSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

## Function/Subroutine Documentation

### subroutine zspsvx (characterFACT, characterUPLO, integerN, integerNRHS, complex*16, dimension( * )AP, complex*16, dimension( * )AFP, integer, dimension( * )IPIV, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( ldx, * )X, integerLDX, double precisionRCOND, double precision, dimension( * )FERR, double precision, dimension( * )BERR, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK, integerINFO)

**ZSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices**

**Purpose:**

```
ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
A = L*D*L**T to compute the solution to a complex system of linear
equations A * X = B, where A is an N-by-N symmetric matrix stored
in packed format 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 diagonal pivoting method is used to factor A as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
2. If some D(i,i)=0, so that D is exactly singular, 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.
```

**Parameters:**

*FACT*

```
FACT is CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= 'F': On entry, AFP and IPIV contain the factored form
of A. AP, AFP and IPIV will not be modified.
= 'N': The matrix A will be copied to AFP and factored.
```

*UPLO*

```
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
```

*N*

```
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
```

*NRHS*

```
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
```

*AP*

```
AP is COMPLEX*16 array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
See below for further details.
```

*AFP*

```
AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
If FACT = 'F', then AFP is an input argument and on entry
contains the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as
a packed triangular matrix in the same storage format as A.
If FACT = 'N', then AFP is an output argument and on exit
contains the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as
a packed triangular matrix in the same storage format as A.
```

*IPIV*

```
IPIV is INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains details of the interchanges and the block structure
of D, as determined by ZSPTRF.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
If FACT = 'N', then IPIV is an output argument and on exit
contains details of the interchanges and the block structure
of D, as determined by ZSPTRF.
```

*B*

```
B is COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
```

*LDB*

```
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
```

*X*

```
X is COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
```

*LDX*

```
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
```

*RCOND*

```
RCOND is DOUBLE PRECISION
The estimate of 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*

```
FERR is 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*

```
BERR is 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*

`WORK is COMPLEX*16 array, dimension (2*N)`

*RWORK*

`RWORK is DOUBLE PRECISION array, dimension (N)`

*INFO*

```
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: D(i,i) is exactly zero. The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. RCOND = 0 is returned.
= N+1: D 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.
```

**Author:**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

April 2012

**Further Details:**

```
The packed storage scheme is illustrated by the following example
when N = 4, UPLO = 'U':
Two-dimensional storage of the symmetric matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = aji)
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
```

Definition at line 277 of file zspsvx.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

zspsvx(3) is an alias of zspsvx.f(3).