# Ritz vectors

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## Parallel processing for Ritz-vector analysis

Prior to beginning Ritz analysis, the machine running ETABS uses multiple cores to factor the stiffness matrix. During Ritz analysis, four main operations include:

- Solving for new vectors
- Cleaning new vectors with respect to previous vectors
- Orthogonalizing the final Ritz vector set
- Post-processing and saving the vectors

Multiple cores are used only for (1), which dominates when the model is large and a fair number of vectors are requested. (2) and (3) are not parallel, and dominate with an increasing number of vectors, growing exponentially with vector quantity. (4) is linear in time with the number of vectors.

Essential behavior is captured more efficiently when using Ritz or Eigen vectors because they reduce the number of degrees of freedom (DOF) within the system. When more than 20-25% of the DOF are needed as Ritz vectors, direct integration should be considered.

## Why does the total number of Ritz vectors requested affect the cumulative modal mass participation ratio for a given number of Ritz vectors?

**Extended Question:** When 1000 Ritz modes are requested, the cumulative modal mass participation ratio for Rx, with 250 modes, is almost 100%. However, when only 500 modes are requested, the cumulative modal mass participation ratio for Rx reaches only about 70% for all 500 modes. Is there an explanation?

**Answer:** Unlike Eigen vectors, Ritz vectors do not always produce the same modes. For example, with sets of 200 and 500 Eigen vectors, the first 200 of each set will be the same. However, with sets of 200 and 500 Ritz vectors, none may be the same, though lower modes tend to converge as the number of modes increases.