F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07GPF (ZPPSVX)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07GPF (ZPPSVX) uses the Cholesky factorization
 $A=UHU or A=LLH$
to compute the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ Hermitian positive definite matrix stored in packed format and $X$ and $B$ are $n$ by $r$ matrices. Error bounds on the solution and a condition estimate are also provided.

## 2  Specification

 SUBROUTINE F07GPF ( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
 INTEGER N, NRHS, LDB, LDX, INFO REAL (KIND=nag_wp) S(*), RCOND, FERR(NRHS), BERR(NRHS), RWORK(N) COMPLEX (KIND=nag_wp) AP(*), AFP(*), B(LDB,*), X(LDX,*), WORK(2*N) CHARACTER(1) FACT, UPLO, EQUED
The routine may be called by its LAPACK name zppsvx.

## 3  Description

F07GPF (ZPPSVX) performs the following steps:
1. If ${\mathbf{FACT}}=\text{'E'}$, real diagonal scaling factors, ${D}_{S}$, are computed to equilibrate the system:
 $DS A DS DS-1 X = DS B .$
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 ${D}_{S}A{D}_{S}$ and $B$ by ${D}_{S}B$.
2. If ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, the Cholesky decomposition is used to factor the matrix $A$ (after equilibration if ${\mathbf{FACT}}=\text{'E'}$) as $A={U}^{\mathrm{H}}U$ if ${\mathbf{UPLO}}=\text{'U'}$ or $A=L{L}^{\mathrm{H}}$ if ${\mathbf{UPLO}}=\text{'L'}$, where $U$ is an upper triangular matrix and $L$ is a lower triangular matrix.
3. If the leading $i$ by $i$ principal minor of $A$ is not positive definite, then the routine returns with ${\mathbf{INFO}}=i$. 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, ${\mathbf{INFO}}=\mathbf{N}+{\mathbf{1}}$ 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 to calculate error bounds and backward error estimates for it.
6. If equilibration was used, the matrix $X$ is premultiplied by ${D}_{S}$ so that it solves the original system before equilibration.

## 4  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

## 5  Parameters

1:     FACT – CHARACTER(1)Input
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.
${\mathbf{FACT}}=\text{'F'}$
AFP contains the factorized form of $A$. If ${\mathbf{EQUED}}=\text{'Y'}$, the matrix $A$ has been equilibrated with scaling factors given by S. AP and AFP will not be modified.
${\mathbf{FACT}}=\text{'N'}$
The matrix $A$ will be copied to AFP and factorized.
${\mathbf{FACT}}=\text{'E'}$
The matrix $A$ will be equilibrated if necessary, then copied to AFP and factorized.
Constraint: ${\mathbf{FACT}}=\text{'F'}$, $\text{'N'}$ or $\text{'E'}$.
2:     UPLO – CHARACTER(1)Input
On entry: if ${\mathbf{UPLO}}=\text{'U'}$, the upper triangle of $A$ is stored.
If ${\mathbf{UPLO}}=\text{'L'}$, the lower triangle of $A$ is stored.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
3:     N – INTEGERInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
4:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
5:     AP($*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}×\left({\mathbf{N}}+1\right)/2\right)$.
On entry: if ${\mathbf{FACT}}=\text{'F'}$ and ${\mathbf{EQUED}}=\text{'Y'}$, AP must contain the equilibrated matrix ${D}_{S}A{D}_{S}$; otherwise, AP must contain the $n$ by $n$ Hermitian matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{AP}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{AP}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
On exit: if ${\mathbf{FACT}}=\text{'F'}$ or $\text{'N'}$, or if ${\mathbf{FACT}}=\text{'E'}$ and ${\mathbf{EQUED}}=\text{'N'}$, AP is not modified.
If ${\mathbf{FACT}}=\text{'E'}$ and ${\mathbf{EQUED}}=\text{'Y'}$, AP is overwritten by ${D}_{S}A{D}_{S}$.
6:     AFP($*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AFP must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}×\left({\mathbf{N}}+1\right)/2\right)$.
On entry: if ${\mathbf{FACT}}=\text{'F'}$, AFP contains the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$, in the same storage format as AP. If ${\mathbf{EQUED}}=\text{'Y'}$, AFP is the factorized form of the equilibrated matrix ${D}_{S}A{D}_{S}$.
On exit: if ${\mathbf{FACT}}=\text{'N'}$ or if ${\mathbf{FACT}}=\text{'E'}$ and ${\mathbf{EQUED}}=\text{'N'}$, AFP returns the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$ of the original matrix $A$.
If ${\mathbf{FACT}}=\text{'E'}$ and ${\mathbf{EQUED}}=\text{'Y'}$, AFP returns the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$ of the equilibrated matrix $A$ (see the description of AP for the form of the equilibrated matrix).
7:     EQUED – CHARACTER(1)Input/Output
On entry: if ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, EQUED need not be set.
If ${\mathbf{FACT}}=\text{'F'}$, EQUED must specify the form of the equilibration that was performed as follows:
• if ${\mathbf{EQUED}}=\text{'N'}$, no equilibration;
• if ${\mathbf{EQUED}}=\text{'Y'}$, equilibration was performed, i.e., $A$ has been replaced by ${D}_{S}A{D}_{S}$.
On exit: if ${\mathbf{FACT}}=\text{'F'}$, EQUED is unchanged from entry.
Otherwise, if no constraints are violated, EQUED specifies the form of the equilibration that was performed as specified above.
Constraint: if ${\mathbf{FACT}}=\text{'F'}$, ${\mathbf{EQUED}}=\text{'N'}$ or $\text{'Y'}$.
8:     S($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array S must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: if ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, S need not be set.
If ${\mathbf{FACT}}=\text{'F'}$ and ${\mathbf{EQUED}}=\text{'Y'}$, S must contain the scale factors, ${D}_{S}$, for $A$; each element of S must be positive.
On exit: if ${\mathbf{FACT}}=\text{'F'}$, S is unchanged from entry.
Otherwise, if no constraints are violated and ${\mathbf{EQUED}}=\text{'Y'}$, S contains the scale factors, ${D}_{S}$, for $A$; each element of S is positive.
9:     B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: if ${\mathbf{EQUED}}=\text{'N'}$, B is not modified.
If ${\mathbf{EQUED}}=\text{'Y'}$, B is overwritten by ${D}_{S}B$.
10:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07GPF (ZPPSVX) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
11:   X(LDX,$*$) – COMPLEX (KIND=nag_wp) arrayOutput
Note: the second dimension of the array X must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, 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 ${\mathbf{EQUED}}=\text{'Y'}$, and the solution to the equilibrated system is ${D}_{S}^{-1}X$.
12:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F07GPF (ZPPSVX) is called.
Constraint: ${\mathbf{LDX}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
13:   RCOND – REAL (KIND=nag_wp)Output
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 ${\mathbf{RCOND}}=1.0/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
14:   FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, an estimate of the forward error bound for each computed solution vector, such that ${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{x}_{j}‖}_{\infty }\le {\mathbf{FERR}}\left(j\right)$ where ${\stackrel{^}{x}}_{j}$ 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.
15:   BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, an estimate of the component-wise relative backward error of each computed solution vector ${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\stackrel{^}{x}}_{j}$ an exact solution).
16:   WORK($2×{\mathbf{N}}$) – COMPLEX (KIND=nag_wp) arrayWorkspace
17:   RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace
18:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If ${\mathbf{INFO}}=i$ and $i\le {\mathbf{N}}$, 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. ${\mathbf{RCOND}}=0.0$ is returned.
${\mathbf{INFO}}={\mathbf{N}}+1$
The triangular matrix $U$ (or $L$) 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.

## 7  Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
• if ${\mathbf{UPLO}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|{U}^{\mathrm{H}}\right|\left|U\right|$;
• if ${\mathbf{UPLO}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|L\right|\left|{L}^{\mathrm{H}}\right|$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. See Section 10.1 of Higham (2002) for further details.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x^∞ ≤ wc condA,x^,b ,$
where $\mathrm{cond}\left(A,\stackrel{^}{x},b\right)={‖\left|{A}^{-1}\right|\left(\left|A\right|\left|\stackrel{^}{x}\right|+\left|b\right|\right)‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. If $\stackrel{^}{x}$ is the $j$th column of $X$, then ${w}_{c}$ is returned in ${\mathbf{BERR}}\left(j\right)$ and a bound on ${‖x-\stackrel{^}{x}‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }$ is returned in ${\mathbf{FERR}}\left(j\right)$. See Section 4.4 of Anderson et al. (1999) for further details.

## 8  Further Comments

The factorization of $A$ requires approximately $\frac{4}{3}{n}^{3}$ floating point operations.
For each right-hand side, computation of the backward error involves a minimum of $16{n}^{2}$ floating point operations. Each step of iterative refinement involves an additional $24{n}^{2}$ operations. 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 equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $8{n}^{2}$ operations.
The real analogue of this routine is F07GBF (DPPSVX).

## 9  Example

This example solves the equations
 $AX=B ,$
where $A$ is the Hermitian positive definite matrix
 $A = 3.23i+0.00 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58i+0.00 -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09i+0.00 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29i+0.00$
and
 $B = 3.93-06.14i 1.48+06.58i 6.17+09.42i 4.65-04.75i -7.17-21.83i -4.91+02.29i 1.99-14.38i 7.64-10.79i .$
Error estimates for the solutions, information on equilibration and an estimate of the reciprocal of the condition number of the scaled matrix $A$ are also output.

### 9.1  Program Text

Program Text (f07gpfe.f90)

### 9.2  Program Data

Program Data (f07gpfe.d)

### 9.3  Program Results

Program Results (f07gpfe.r)