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# holmberg1

## PURPOSE

Computes moments for higher order reliability methods.

## SYNOPSIS

E=holmberg1(S,a,b,Q1)

## DESCRIPTION

```  HOLMBERG1 Computes moments for higher order reliability methods.

CALL: E=holmberg1(S,a,b,Q1);

Computation of the expectation of
sqrt(pi/2)*(b'*X)*(X'*Q*X)*(2*normcdf(a'*X)-1)
if S is normally distributed with mean zero and covariance matrix
S.```

## CROSS-REFERENCE INFORMATION

This function calls:
 trace Sum of diagonal elements.
This function is called by:
 rqlf_asympt Gives first two terms in an asymptotic expansion of the

## SOURCE CODE

```01 function E=holmberg1(S,a,b,Q1)
02 % HOLMBERG1 Computes moments for higher order reliability methods.
03 %
04 % CALL: E=holmberg1(S,a,b,Q1);
05 %
06 % Computation of the expectation of
07 % sqrt(pi/2)*(b'*X)*(X'*Q*X)*(2*normcdf(a'*X)-1)
08 % if S is normally distributed with mean zero and covariance matrix
09 % S.
10 Q1=(Q1+Q1')/2;
11 term1=a'*S*b*trace(S*Q1);
12 term2=2*a'*S*Q1*S*b;
13 term3=-(a'*S*b)*(a'*S*Q1*S*a)/(1+a'*S*a);
14 E=(term1+term2+term3)/sqrt(1+a'*S*a);
15```

Mathematical Statistics
Centre for Mathematical Sciences
Lund University with Lund Institute of Technology

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