* Ziems . (2022) ziems-etal-2022-moralZiems, C., Yu, J., Wang, YC., Halevy, A. Yang, D.  202205\.  The Moral Integrity Corpus: A benchmark for ethical dialogue systems The moral integrity corpus: A benchmark for ethical dialogue systems.  Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) Proceedings of the 60th annual meeting of the association for computational linguistics (volume 1: Long papers) ( 3755–3773).  Dublin, IrelandAssociation for Computational Linguistics.  https://aclanthology.org/2022.acl-long.261 10.18653/v1/2022.acl-long.261

An overview of the annotation procedure is given in §A, followed by a detailed
description of the first (§B) and second (§C) annotation round.

##  Appendix A Annotation Guidelines: Procedure

The annotation procedure is split in two rounds.

For a given sentence and the title of the blog post the sentence was taken
from (= extra context)

Annotation Round 1 (§B)

  1. 1.

Mark the people and groups of people mentioned in the sentence (highlight
people entities in sentence);

  2. 2.

Indicate whether or not the author implies a character trait for at least one
of the marked entities (radio button, yes/no);

Only sentences for which the annotators answered ‘yes’ for step 2 are sent to
annotation round 2.

Annotation Round 2 (§C)

  1. 1.

Indicate your attitude as a reader towards the writer (slider, five-point
Likert scale);

  2. 2.

Reformulate the sentence such that it reflects what the writer ‘basically
says’ (free text);

For each entity marked during step 1.1 of ROUND 1:

  1. 1.

Indicate whether or not the author implies a character trait of the given
entity (dropdown list, yes/no).

If yes, continue with (a), (b), and (c). If no, submit annotation.

    1. (a)

Describe the implied character trait (free text);

    2. (b)

Label whether society regards the trait as good or bad (dropdown menu,
Good/Bad);

    3. (c)

Label the Sphere of Action the character trait belongs to (dropdown menu, 10
class labels) and label the degree of appropriateness the character trait
belongs to (dropdown menu, 3 class labels).

##  Appendix B Annotation Guidelines: Round 1

In the first annotation round, annotators need to identify the people entities
mentioned in the sentences by highlighting them in the sentence and then
indicate whether or not the author of the sentence seems to imply a character
trait of at least one people entity marked in the sentence. It is made clear
in the interface of the first annotation round that the annotators need to
fulfill two tasks, marked by (1/2) and (2/2). The annotators are unable to
submit the task without answering the second question to avoid incomplete task
forms. Candidate annotators had to pass a qualification test to have access to
the annotation round.

Figure 8: Annotation interface for annotation round 1. Annotators perform two
tasks: Mark the entities in the given sentence (${sentence}$) using the label
provided on the right (People, green label) and indicate whether the author of
the sentence implies a character trait of at least one of the marked entities.
The title of the blog post (${title}$) from which the sentence was taken is
given as context surrounding the sentence.

###  B.1 Task 1: Mark People Entities in a Sentence

The annotators mark the people and groups of people mentioned in the sentence
(Figure 8). If there are no entities to label, they can tick the ‘No entities
to label’ box at the bottom right of the interface and submit the task form.
The title of the blog post from which the sentence was taken is given below
the sentence to contextualize its content. The general rules for marking
people entities are given below the sentence and title such that the
annotators were reminded of them when performing the task. We provide a more
in-depth discussion of the rules with illustrative examples under the
instructions tab at the top left corner of the interface.

Figure 9: The instructions for marking the people entities in the given
sentence can be opened when clicking on the instructions tab at the top left.
They give a few short example sentences and list which references should not
be highlighted. Additional information about which references to highlight and
how much of the reference needs to be highlighted can be accessed by clicking
on the clickable blue sentences in the instructions.

((a)) Pop-up screen when clicking on the instructions link “How much to
highlight?”

((b)) Pop-up screens when clicking on the instructions link “Which references
do you highlight?”

Figure 10: The pop-up instruction when clicking on the clickable blue links
provided in the instructions.

####  B.1.1 Instructions: What to Mark

  * •

Mark words and phrases that refer to a person or a group of people Only mark
words that refer to people or a group of people such as an organisation.
Examples:

    * –

They welcome refugees from the Ukraine.

    * –

I saw this comment from some people in the US.

    * –

He talked about his friend Chad.

    * –

I’ve liked three different guys that were the older brother.

    * –

I have friends who are Trump supporters.

    * –

Past conservatives would not have worried about this.

    * –

NATO should be taking more responsibility.

  * •

Mark only one reference per person/group in each sentence If there are more
than one instance that refers to the same person or group, mark the most
descriptive reference or, in case they are identical, the first one.

    * –

Cheerleaders like to dance and they love to cheer. (both ‘cheerleaders’ and
‘they’ refer to the same group - mark the most descriptive instance).

    * –

They like to dance but they hate to sing. (both instances of ‘they’ refer to
the same group - mark the first instance).

  * •

Mark the most descriptive reference to a person or group In case the same
entity is referred to more than once in the sentence, mark the reference that
is most descriptive. Example:

    * –

Cheerleaders like to dance and they love to cheer. (Mark ‘cheerleaders’ as it
is more descriptive than ‘they’)

  * •

Do not mark animals or objects Example:

    * –

Our dog Ziggy is one year old. (‘Our dog Ziggy’ is not a person).

  * •

Do not mark the author (‘I’) Example:

    * –

I saw this comment from some people in the US. (‘I’ refers to the narrator)

  * •

Do not mark void, vague, or too general references Examples:

    * –

Void ‘you’/‘we’. Example: If you would compare the two objects. (‘you’ does
not refer to a specific person; void); Yet, we are seeing mask mandates pop up
around the country. (‘we’ does not refer to a specific group; void).

    * –

Vague. Example: I don’t mean to say that certain people with particular
sibling orders have less empathy, but I’ve noticed some key differences in
romantic relationships. (‘certain people with particular sibling orders’ is
too vague); For me to treat someone as fully human,… (‘someone’ is too vague)

    * –

Too general. Example: People say that it will be better. (‘people’ does not
refer to a specific group of people).

  * •

Do not mark predicates Predicates are often preceded by verbs such as ‘to be’.
Example:

    * –

Two of these guys are the types of dudes that hook up with girls. (‘the types
of dudes that hook up with girls’ is predicate to ‘two of these guys’)

  * •

Do not mark references to a person/group that are part of phrase referring to
other people/groups Example:

    * –

I have friends who are Trump supporters (‘Trump’ is part of another reference,
namely ‘friends who are Trump supporters’)

    * –

Joe Biden’s top covid advisor was on CNN today. (‘Joe Biden’ is part of ‘Joe
Biden’s top covid advisor’)

####  B.1.2 Instructions: How Much to Mark

  * •

Mark all the words that specify the person referred to Example:

    * –

They welcome refugees from the Ukraine. (marking only ‘refugees’ would be
incomplete as it then refers to a larger, more general group).

    * –

I saw this comment from some people in the US. (including ‘some’ and ‘in the
US’ is more specific than leaving one of the two, or both, out).

    * –

He talked about his friend Chad. (including ‘his friend’ provides more
information on the relation between ‘he’ and ‘Chad’).

    * –

I’ve liked three different guys that were the older brother. (including the
relative clause ‘that were the older brother’ specifies the identity of the
guys).

    * –

I have friends who are Trump supporters. (including the relative clause ‘who
are Trump supporters’ specifies the identity of the writer’s friends).

    * –

Past conservatives would not have worried about this. (including adjective
‘past’ is necessary to refer to the right group of conservatives).

  * •

Except

    * –

Adjectives preceding the person/group noun that denote their
personality/character traits; they can be removed without changing the person
or group they refer to.

      * *

Very wise tradesmen or dumb trust fund babies that somehow make their way
through uni. (‘very wise’ and ‘dumb’ describe character traits).

      * *

Although I’m sure that there are people who never go to university that are
far smarter than I am, on the whole it would be far more effective for only
reasonable intelligent and conscientious people (people $\Rightarrow$ people
who went to university) to decide on the future of our society. (‘reasonable
intelligent and conscientious’ describe character traits).

    * –

Relative clauses that do not specify the kind of person/group is referred to;
leaving them out would not change the reference.

      * *

There are some online pseudo-“leftists" like Jimmy Dore who willingly spread
propaganda. (‘who willingly spread propaganda’ does not specify the
reference.)

###  B.2 Task 2: Indicate Presence or Absence of Implied Character Trait

After labeling the entities in the sentence, the annotators now indicate
whether or not the author of the sentence seems to imply a character trait of
at least one entity they have marked in the sentence.

####  B.2.1 Instructions: Definition of Character Trait

We provide the annotators with a definition of character trait in the
instructions, which can be accessed by clicking on the instructions tab on the
top left corner.

((a)) Annotator instructions regarding Task 2.

((b)) Pop-up screen when clicking on the instructions link “What is a
character trait?”.

Figure 11: Instructions regarding Task 2 (a) and definition of character trait
(b)

A character trait describes an aspect of a person’s behavior or attitude. The
set of character traits then forms someone’s personality. A trait is also
voluntary so it is not forced on that person by someone or something. A person
has control over his/her character traits. A person’s character traits cause
him/her to act a certain way. Character traits can very often be described
using descriptive adjectives such as honest, sincere, brave, loyal, vain,
timid, and calm. Examples:

  * •

Eric disguises himself. (Eric is sneaky).

  * •

Eric likes to talk about himself. (Eric is vain).

  * •

Eric does not want to spend much money. (Eric is stingy).

####  B.2.2 Instructions: The Author Implies Trait of at Least One Entity

The annotators should ask themselves whether the author seems to suggest
something about the character of the people in the sentence. It is important
to distinguish between character traits implied by the author, showing that
the author seems to hold certain beliefs about people, and character traits
that a reader or other people mentioned by the writer belief people have (a
reader or mentioned person holds certain beliefs about people). Like any other
person, an annotator can have certain, pre-existing beliefs about (groups of)
people or can be aware of pre-existing beliefs that others have about them. We
therefore stress in the phrasing of the second task that the annotators should
think about what the author seems to imply by underlining the author: “Does
the author seem to imply a character trait of at least one ‘People’-labeled
person/group?”. We also underline at least one to stress that a character
trait for all entities need to be implied for indicating ‘yes’.

###  B.3 Qualification Test: Round 1

Candidate annotators first have to pass a qualification test before they can
start the annotation round. They were automatically admitted to the annotation
round if they answered all questions in the qualification test correctly. The
qualification test instructs the annotators on how to adequately mark people
entities in a sentence by going over five main rules. For each rule, the
annotator is presented with an example sentence. They then need to pick the
correct labeling answer from the options below the sentence (radio button). We
indicate the correct answer with (X).

Test description: We go over FIVE labeling rules for this task. For each rule,
you will be presented with an example sentence and you need to pick the
correct labeling answer from the options below. If you pick all the correct
answers, you can automatically start the labeling task.

  * •

Rule 1: Highlight all words, phrases, or sections of the sentence that refer
to people other than the author (i.e., ’I’).

    * –

Sentence: "I talked with my friend Chad about some men from Texas and people
who like Star Wars."

      * *

I - Chad - men - people

      * *

(X) my friend Chad - some men from Texas - people who like Star Wars

  * •

Rule 2: Highlight only one reference per person/group + choose most
descriptive reference.

    * –

Sentence: "Cheerleaders like to dance and they love to cheer."

      * *

(X) Cheerleaders

      * *

Cheerleaders - they

      * *

they

  * •

Rule 3: Highlight all words specifying the reference; except adjectives
indicating personality (e.g., reasonable, vain).

    * –

Sentence: "The blond girl from Canada told that there are honest girls and
generous boys in her class."

      * *

girl - girls - boys

      * *

The blond girl from Canada - honest girls - generous boys

      * *

(X) The blond girl from Canada - girls - boys

  * •

Rule 4: Do not highlight vague, void and too general references.

    * –

Sentence: "If you meet someone online, people will tell you to be cautious"

      * *

you - someone - people

      * *

(X) [None] (you = void, someone = vague, people = too general)

  * •

Rule 5: Do not highlight predicates (often after a modal verb like ’to be’:
’Person is [predicate]’) and people that are part of another, larger reference

    * –

Sentence: "Barack Obama became the US president when he was 48 and Barack
Obama’s wife was the First Lady"

      * *

(X) Barack Obama - Barack Obama’s wife

      * *

Barack Obama - the US president - wife - the First Lady

##  Appendix C Annotation Guidelines: Round 2

The annotators start by reading the sentence (${sentence}) and the blog post
from which the sentence is taken (${title}). They are then asked to describe
their attitude towards the author and formulate their interpretation of the
given sentence. Going over all people identified in the sentence, they
indicate for each entity whether or not the author of the sentence implies a
character trait of that entity. For all entities with an implied character
trait, the annotators also describe the trait, label the evaluation of that
trait in society, and classify it in Virtue Ethics by labeling the Sphere of
Action the trait belongs to and its contextual appropriateness. It is made
clear in the interface (Figure 12) that the annotators need to fulfill four
tasks, marked by (1/4) to (4/4). They can access the instructions with
illustrative examples at any time during annotation by clicking on the
instructions tab at the top left corner of the interface (Figure 13). The
instructions then appear on the left and can stay visible during the
annotation process. They disappear when clicking on the instructions tab
again. The annotators are unable to submit the task form without providing an
interpretation, which is the most important part of this annotation round.
Candidate annotators had to pass a qualification test to have access to the
annotation round.

Figure 12: The main annotation interface of ANNOTATION ROUND 2.

Figure 13: Overview of the instructions that appear when clicking on the
instructions tab at the top left of the annotation interface.

###  C.1 Task 1: Describe Attitude Towards Author of Sentence

The annotators describe their attitude towards the author of the sentence upon
reading the sentence. We stress in the task description that they should
describe their own attitude by underlining your attitude in the task
description and state in brackets that their attitude is their impression of
the author based on the sentence. They change the slider given below the task
description and set it to the best-fitting attitude on a five-point Likert
scale ranging from very negative tot very positive. When moving the slider,
the value to which they set it changes to blue and depicts the number
referring to the attitude they set it to (Figure 14).

Figure 14: Attitude slider shows attitude values on a five point Likert scale.

####  C.1.1 Instructions: Definitions of Attitude Scale

We provide the annotators with a definition of each attitude type in the
instructions. They can access the overview of attitude definitions by clicking
on the clickable blue sentence “What do the attitude mean?” in the
instructions. The overview then appears as a pop-up screen.

  * •

Very negative After reading the sentence, the annotator has a very negative
view of the author. They completely disagree with what the author said. They
would never say or think the same, or never phrase it like the author did.

  * •

Negative After reading the sentence, the annotator has a negative view of the
author. They disagree with what the author said. They would not say or think
the same, or phrase it like the author did.

  * •

Neutral After reading the sentence, the annotator does not have a negative of
positive view of the author. They do not agree of disagree with what the
author said.

  * •

Positive After reading the sentence, the annotator has a positive view of the
author. They agree with what the author said. They would say or think the
same, or phrase it like the author did.

  * •

Very positive After reading the sentence, the annotator has a very positive
view of the author. They completely agree with what the author said. They
would definitely say or think the same, or phrase it like the author did.

Figure 15: Overview of the different attitude types and their description
given in the instructions.

###  C.2 Task 2: Formulate Interpretation of Sentence

The annotators write their interpretation of the given sentence in their own
words. We stress that they should formulate their own interpretation by
underlining your interpretation in the task description. They are also
explicitly asked to not simply copy the sentence. The instructions
additionally state that your interpretation may reveal implicit information
contained within the sentence.

####  C.2.1 Instructions: Examples of Interpretations

We provide the annotators with several examples of possible interpretations
for an example sentence, which can be accessed by clicking on the clickable
blue phrase “Examples interpretations” in the instructions (Figure 16). The
examples appear in a pop-up screen. We phrase the interpretations in such a
way so that the underlying messages and hidden moral judgments surface.

Given sentence: “They need to force the unemployed to look for work."

  * •

Possible interpretation (1): “As the unemployed are too lazy to look for a
job, we need to force them to do this.”

  * •

Possible interpretation (2): “They, who are superior, need to force the
unemployed to look for work.”

  * •

Possible interpretation (3): “They need to force the unemployed to look for
work, because they are superior to them and the unemployed are too lazy to do
it themselves.”

Figure 16: Examples of possible interpretations of a given sentence given in
the instructions.

###  C.3 Task 3: Categorize Implied Character Trait for Each Entity in
Sentence

The annotators need to answer the following questions for each entity that was
previously marked in the first annotation round:

  * •

Does the author seem to imply a character trait of this entity? (Task 3.1)

  * •

If Task 3.1 = yes:

    * –

What is the character trait? (Task 3.2)

    * –

How does society evaluate this character trait? (Task 3.3)

    * –

Where is the character positioned in Virtue Ethics in terms of Sphere of
Action and contextual appropriateness? (Task 3.4)

The interface presents the classification table for classifying the character
traits in Virtue Ethics right below the task description (Figure 17(a)). The
annotators are then given the list of people entities for which they need to
provide annotations (${people}). The interface then contains entity annotation
blocks, with the number of blocks equal to the number of entities in the
people entities list (Figure 17(b)). Each block first presents the sentence
such that the annotator does not need to scroll up to read the sentence again.
The block then contains Task 3.1, 3.2, 3.3, and 3.4. The tasks are framed as
sentences, and the annotators need to select the labels from drop-down lists.

((a)) Virtue Ethics classification table.

((b)) Entity annotation block for the first people entity (${p1}) in the
people entities list (${people}).

Figure 17: Interface for tackling Task 3.

####  C.3.1 Task 3.1: Indicate Presence Implied Character Trait of Entity

The annotators need to indicate whether or not the author implies a character
trait of the given people entity. They do this by choosing between yes and no
from a drop-down list (Figure 18). If they indicate yes, they need to continue
with the annotation for this entity. This is also stated in bold after the
drop-down list. Otherwise, they can continue to annotate the next entity or,
in case this entity is the last one, submit the task form.

Figure 18: Drop-down list for solving Task 3.1.

#####  Instructions: Definition of Character Trait

The annotators are given the same definition of character trait given in the
first annotation round, see § B.2.1. The definition can be accessed by
clicking on the clickable blue sentence “What is a character trait?” in the
instructions, which then appears in a pop-up screen.

####  C.3.2 Task 3.2: Identify Implied Character Trait of Entity

The annotators describe the implied character trait in their own words in the
free-text text box (Figure 19).

Figure 19: Input for Task 3.2. is provided as free text.

#####  Instructions: How to Describe Trait

The annotators are encouraged to describe the trait using an adjective
(“Describe the trait using an adjective (e.g., honest, vain) here.”) and not
to use an adjective that is presented in the Virtue Ethics classification
table given above (“Try using a term that is not in the table able.”), see
Figure 19. When the annotators start describing the trait, the two
instructions appear above the free-text box.

####  C.3.3 Task 3.3: Evaluate Implied Character Trait of Entity in Society

The annotators decide whether society thinks this is a good (positive) or bad
(negative) trait (Figure 20). Or in other words, does society praise people
for having this trait or frown upon or even reject this trait, respectively.

Figure 20: Drop-down list for solving Task 3.3.

#####  Instructions: Focus on What Society Thinks

Since we are interested in how the character trait is evaluated in society, we
mark society in bold in the task description (“Society thinks this trait is”).
This way, we encourage annotators to think beyond their own evaluations of
character trait and position the trait in the society they are part of.

####  C.3.4 Task 3.4: Classify Implied Character Trait of Entity in Virtue
Ethics

The annotators classify the implied character trait in Virtue Ethics by
selecting the best-fitting labels from the drop-down lists (Figure 22). A
table containing all the Virtue Ethics labels is given below the task
description so that the annotators can easily consult the possible labels.
They first classify the trait in a Sphere of Action (table rows) and then by
its contextual appropriateness (table columns). When clicking on the
instructions tab at the top left corner, the annotators see an overview of all
the Spheres of Action with a brief description for each Sphere of Action.

Before starting the annotation task, candidate annotators had to pass an
instruction test in which the Virtue Ethics and its labels were explained and
participants had to select the appropriate contextual appropriateness label
for ten sentences, with each sentence related to a different Sphere of Action.
The annotators were allowed to start the annotation task if they correctly
labeled nine sentences.

((a)) Drop-down list for selecting Sphere of Action labels.

((b)) Drop-down list for selecting contextual appropriateness labels.

Figure 21: Drop-down lists for solving Task 3.4.

((a)) Definition of Giving (and taking) - small amounts of money.

((b)) Definition of Confidence (fear, uncertainty).

Figure 22: Definitions of two Spheres of Action that appear when clicking on
the clickable blue phrases 1-confidence (fear, uncertainty) (a) and 3-giving
(and taking - small amounts of money (b).

#####  Instructions: Virtue Ethics

We adopt the well-established Aristotelian Ethics (Hursthouse, 1999), more
specifically Nichomachean Ethics, to label people’s moral virtues and vices.
Aristotle argued that a virtue/vice is a character trait or a state of
someone’s character/personality. A Virtue of trait causes a person to act in a
manner that is appropriate and desired by society in a given context. It is
not because someone acts in a Virtue of manner that that person is actually
virtuous. It is important that the action is voluntary and within the control
of the individual. By contrast, a vicious trait causes a person to act in a
manner that is frowned upon, disapproved or even forbidden by society.

Aristotle identified several Spheres of Action (SoA) that cause people to act
in various ways. Those different kinds of actions are put on an axis according
to their degree or contextual appropriateness: deficiency, mean and excess.
Aristotle stated that a Virtue of person acts in a moderate and contextually
appropriate manner (= mean), while a morally faulty person resides to morally
deficient (= deficiency) or exceeding actions (= excess). We therefore speak
of Vice of Deficiency, Virtue of Mean, and Vice of Excess. Table 8 presents an
overview of the Spheres of Action and the contextual appropriateness labels.
We take the descriptions of the SoAs from Hursthouse (1999).

SoA or Feeling |  Vice of Deficiency |  Virtue of Mean |  Vice of Excess
---|---|---|---
Confidence (fear, uncertainty) |  cowardice |  courage |  rashness
Pleasures of the body |  insensibility |  temperance |  profligacy
Giving (and taking) - small amounts of money |  stinginess |  liberality, generosity |  prodigality
Giving (and taking) - other, added value |  meanness |  magnificence |  vulgarity
Pride (honour = cause) |  little-mindedness |  high-mindedness |  vanity
Ambition (honour = goal) |  lack of ambition |  proper ambition |  over-ambitiousness
Anger |  easy-tempered, spiritlessness |  gentleness |  hard-tempered, wrathfulness
Pleasure (and pain) of others |  cross, contentious |  agreeableness, friendliness |  flattery
Truth, honesty about oneself |  irony |  truthfulness |  boastfulness
Amusing conversation |  boorishness |  wittiness |  buffoonery
Table 8: Overview of Spheres of Actions (SoA) with their the virtue and vices.

##### Confidence (Fear, Uncertainty)

How someone deals with fearful, uncertain events/situations and to which
extent someone is confident about their actions.

  * •

Cowardice [Vice of Deficiency] Lack of confidence, excess in fear
Someone who has too much fear when dealing with uncertain situations when
dealing with uncertain situations. This person also fears the wrong things and
deals with things in the wrong way. It could be said that this persons is
deficient in confidence when being faced with situations that excite fear and
uncertainty.

  * •

Courage [Virtue of Mean] Appropriate fear and confidence
Someone who fears the right situations and the level of fear depends on the
fear-exciting nature of each of those situations (fearful things differ in
importance and thus a courageous person fears it accordingly). This person
deals with fear-exciting situations with appropriate courage/confidence, in
the appropriate way, and at the appropriate time. This person’s courage
originates from a sense of nobility and is not excited under compulsion
because being courageous is a voluntary trait.

  * •

Rashness [Vice of Excess] Lack of fear, excess in confidence
Someone has too little fear and too much confidence when dealing with
uncertain situations. This person pretends to have a courage (s)he has not.
The high level of confidence is often caused by a sense of superiority and
invincibility.

##### Pleasures of the Body

How someone deals with and is able to control and fulfil the needs and
pleasures of the body.

  * •

Insensibility [Vice of Deficiency] Too little delight in bodily pleasures
Someone who takes less delight in such pleasures than ought to. This is rather
uncommon, almost non-existent as these urges are innate and attached to our
animal nature.

  * •

Temperance [Virtue of Mean] Self-control, appropriate desire to fulfill bodily
pleasures
Someone who has temperance and self-control. This person desires moderately,
in the appropriate way, at the appropriate time and at the appropriate place.
The absence of these pleasant things or the abstinence from them is not
painful to a temperate person.

  * •

Profligacy [Vice of Excess] Too much delight in bodily pleasures
Someone who is a slave to bodily pleasures and loves these delights/pleasures
too much. For example, heavy drinking (drinking more than is good for you),
obesity, and gluttony. This person also enjoys things that (s)he should not
enjoy, in the wrong manner, and more than necessary. The life of a profligate
person is led by his/her appetites. Other examples: adultery and outrage.
Example:

    * –

There was one guy who had a younger brother. He ($\Rightarrow$ one guy who had
a younger brother), ahem, tried to take advantage of me when I was drunk.

##### Giving (and Taking) - Small Amounts of Money

How and to which someone spends their resources (monetary and non-monetary)

  * •

Stinginess [Vice of Deficiency] Lack of giving, excess in taking
Someone who cares more for wealth than is right, is fond of money. This person
takes more than (s)he gives. Examples: greed, meanness, lending money at high
interests. The mean person never does anything without thinking twice and
always considers how things can be done at the least possible cost.

  * •

Liberality, Generosity [Virtue of Mean] Spending resources to create added
value for others
Someone who spends their money and other resources on proper things, allocate
resources to the right people (generosity), acquires wealth by taking from the
proper sources, and desires to reach goals in the right manner (proper amount
spent). The liberal person invests in public objects and other people to
create added value, which is not always valued in money. This person does go
for the cheapest or most expensive way to do things (focus on means), but
instead focuses on the goal.
Example:

    * –

However, he has a history of charity that goes far beyond this instance. He
invested over one billion in a poor region in Russia, transforming the local
economy and greatly raising the standard of living there. (he)

  * •

Prodigality [Vice of Excess] Excess in giving (lack in taking)
Someone who squanders money in riotous, wasteful living. The wasteful person
wastes money on things that do not create added value to others (improper
objects), or spends too much on small objects. This person also likes to
display their wealth to others.

##### Giving (and Taking) - Other, Added Value

How someone spends their wealth to create added value and to which things
(their worth is not always measurable in terms of money), spending/giving
according to their wealth.

  * •

Meanness [Vice of Deficiency] Lack of giving, excess in taking
The mean person likes to do things at the lowest possible cost.

  * •

Magnificence [Virtue of Mean] Spending resources to create added value for
others
The magnificent person invests in public objects and other people to create
added value, which is not always valued in money. This person does not go for
the cheapest or most expensive way to do things (focus on means), but instead
focuses on the goal.

  * •

Prodigality, Wastefulness [Vice of Excess] Excess in giving, lack in taking
Someone who squanders money in riotous, wasteful living. The wasteful person
wastes money on things that do not create added value to others (improper
objects), or spends too much on small objects. This person also likes to
display their wealth to others.

##### Pride (Honour = Cause)

How someone communicates about and acts according to one’s honour/pride.

  * •

Little-mindedness [Vice of Deficiency] Lacks pride, false modesty
Someone who claims less than (s)he deserves and deprives him/herself of what
(s)he deserves. This should not be confused with modesty as this person
actively deprives himself of honour (false modesty). This person is too
retiring.

  * •

High-mindedness [Virtue of Mean] Proper sense and display of honour
Someone who claims much but also deserves much. This differs from modesty as a
modest person has little to claim about - high-mindedness implies some kind of
greatness. The high-minded person also does not seek honour from just everyone
(does not look for praise) or on trivial grounds. This person is not easily
moved to admiration, does not gossip or speak evil of others. Brooding on the
past or blaming others is not something a high-minded person would do.

  * •

Vanity [Vice of Excess] Excess in pride, loves oneself
Someone who claims much without deserving it. The vain person considers
him/herself the better person (sense of superiority) and believes that (s)he
deserves greater things. This sense of superiority is unjust and unjustified.
This person is ignorant about his/her honour, likes to talk about his/herself,
looks down on others and speaks evil of others. At a professional level, they
tend to take positions for which they are unfit.
Example:

    * –

There are very low quality psychologists who go around trying to tell people
what to do and fix their problems but they are not qualified because they have
studied a shell major, which is psychology.

##### Ambition (Honour = Goal)

How someone builds towards more honour, how someone works on becoming a more
Virtue of person.

  * •

Lack of ambition [Vice of Deficiency] Lacks ambition
Someone who has little desire to improve one’s character or condition.
Example: laziness. This person also tends to blame others for his/her lack of
ambition.
Example:

    * –

Psychologist is just someone who cannot man up and study medicine so they go
for the sad mockery (…) People ($\Rightarrow$ psychologists) study it as a
wild card/placeholder major. (psychologists)

  * •

Proper ambition [Virtue of Mean] Healthy ambition
Someone who is ambitious in the proper manner, in the proper situations and
for reaching appropriate goals.

  * •

Over-ambitiousness [Vice of Excess] Too much ambition
Someone who desires too much - often at the expense of others. For this
person, the goal justifies the means.

##### Anger

How someone deals with emotions, especially negative emotions such as anger.

  * •

Easy-tempered, spiritlessness [Vice of Deficiency] Too little anger
Someone who does not sufficiently deal with their negative emotions, quite
emotionless. The easy-tempered person also lacks the spirit to defend
him/herself and, more importantly, others.

  * •

Gentleness [Virtue of Mean] Deals with negative emotions such as anger in a
proper way.
Someone who is angry on the right occasions, with the right persons, in the
right manner, for the right reasons and for the right length of time. The
gentle person does not lose his/her balance and does not get carried away by
emotions. This person is ready to forgive and is not eager to take vengeance.

  * •

Hard-tempered, wrathfulness [Vice of Excess] Too much anger
Someone who is easily angered by anything and anyone on any occasion. This
hard-tempered person’s bursts of emotions can be either very short or very
long (vengeance and punishment). This person is also prone to violence,
vengeance and punishment. (S)he does not easily forgive. The extremes are
choleric people.
Example:

    * –

I understand Putin is insane. (Putin)

##### Pleasure (and Pain) of Others

How someone agrees (pleasure of other) or disagrees (pain of other) with
his/her interlocutor in any type of interaction.

  * •

Cross, Contentious [Vice of Deficiency] too much pain of other, disagreement
Someone who objects to and disagrees with everything and anyone. The cross
person does not consider the (emotional) pain they give to people, sets
his/her face against everything.
Example:

    * –

“My mother took some pictures and posted them on Facebook, where they were
quickly filled with horrified comments from various Americans who she is
friends with." (various Americans who she is friends with)

  * •

Agreeableness, friendliness [Virtue of Mean] appropriate mix of agreement and
disagreement
Someone who (dis)agrees when she oughts to and not because of love or hate.
The agreeable person fits their behaviour to the context and the person they
are talking to.

  * •

Flattery [Vice of Excess] Too much pleasure of other, agreement
Someone who pleases people by praising everything and not objecting to
anything. This “people-pleaser" feels the need to avoid discomfort and
maximize the pleasure of others. When this person has a goal that leads to
this pleasing, then we call it flattery.

##### Truth, Honesty About Oneself

How truthful someone is in their interaction with others: what they say and
how they say it.

  * •

Irony [Vice of Deficiency] Minimizing truthfulness by disclaiming and
depreciating the truth
Someone who disclaims what (s)he has or depreciates it. This depreciatory way
of speaking allows him/her to avoid parade. The character of ironic person
should not be confused with modesty and irony (as a figure of speech).

  * •

Truthfulness [Virtue of Mean] truthful in word and deed
Someone who is truthful in word and deed. The truthful person never
exaggerates or diminishes the truth, and shuns falsehood as a base thing (does
not take pleasure in it). This person is more inclined towards understatement
than overstatement of the truth.
Example:

    * –

I also understand more thoughtful conservatives who voted for Trump, because
regardless of their opinions on him ($\Rightarrow$ Trump) as a person, they
knew he would support conservative causes, which he did. (Trump)

  * •

Boastfulness [Vice of Excess] Pretension and twisting the truth
Someone who likes to pretend to be someone they are not or pretend to hold
ideas that others esteem even though they do not. Examples: liars and
hypocrites.
Example:

    * –

I will make it very clear, I’m not pro Russian or pro Ukraine, I think both
sides are to be blamed for this war and mostly we are seeing only one side of
the story from the western media. (the western media)

##### Amusing Conversation

How someone behaves during amusing conversation and mixing with others

  * •

Boorishness [Vice of Deficiency] Too boring and serious
Someone who never says anything laughable, contributes little to the
conversation, is too serious and takes everything in ill part.

  * •

Wittiness [Virtue of Mean] Graceful jest and tact
Someone who has a lot of tact, jests gracefully, does not ridicule anyone.

  * •

Buffoonery [Vice of Excess] Excess in ridicule
Someone who does ridicule things just for a laugh, not elegant in their wit,
easily resides to offense. The ridicule person likes to make jests of people,
vilify them or laugh with them.

###  C.4 Qualification Test: Round 2

Candidate annotators first have to pass a qualification test before they can
start the annotation round. They were automatically admitted to the annotation
round if they answered all questions or all but one question in the
qualification test correctly. The qualification test starts by briefly
explaining the three contextual appropriateness labels, called behaviour types
in the test for simplicity, after which the annotators have to select the
correct description of the types (radio button). It then goes over all Spheres
of Action, called situations in the test for simplicity. An example sentence
and a person of interest is given for each situation and the annotators are
expected to select the best-fitting behaviour type for the person of interest
as implied in the sentence (radio button). The sentences present rather clear
examples of the behaviour type. We indicate the correct answer with (X).

Test description: In this project, you will be asked to describe IMPLIED (=
indirectly suggested) CHARACTER TRAITS and classify them given a predefined
set of situations and behaviors. We go over the ten different SITUATIONS (=
spheres of action), each having three BEHAVIOR types (’vice of deficiency’,
’virtue of mean’, ’vice of excess’).

(3 BEHAVIOR TYPES) Each situation incites certain behaviors. Those behaviors
can either be desired/positive (= virtue) or undesired/negative (= vice).

\- A desired behavior (= virtue) is APPROPRIATE to the situation and people
involved. It is often a moderate behavior. We therefore classify this type of
desired behavior as ’VIRTUE OF MEAN’.

\- An undesired behavior (= vice) is disproportionate to the situation and
people involved. If it is EXCESSIVE, or ’too much’, we call this type of
undesired behavior as ’VICE OF EXCESS’. The behavior can also be DEFICIENT, or
’too little’. We then call this undesired behavior ’VICE OF DEFICIENCY’.

  * •

Pick the correct description for the three behavior types: VICE OF DEFICIENCY
- VIRTUE OF MEAN - VICE OF EXCESS.

    * –

"too much" - appropriate - "too little"

    * –

(X) "too little" - appropriate - "too much"

(10 SITUATIONS) For each situation (S), we give an example sentence and a
person of interest (– Person). You need to pick the BEST-FITTING behavior type
for the person of interest, as IMPLIED in the sentence.

  * •

(S1) CONFIDENCE (FEAR, UNCERTAINTY): how someone deals with fear-inciting,
uncertain events and the extent to which someone is confident about their
actions.

    * –

"They made increasingly reckless investments." – They

      * *

Vice of Deficiency $\Rightarrow$ cowardice, or "too little" confidence (= too
much fear/uncertainty)

      * *

Virtue of Mean $\Rightarrow$ courage, or "appropriate" confidence, fear and
uncertainty

      * *

(X) Vice of Excess $\Rightarrow$ rashness, or "too much" confidence (= too
little fear/uncertainty)

  * •

(S2) PLEASURES OF THE BODY: how someone deals with and is able to control and
fulfill needs and pleasures, especially those of the body.

    * –

"He, ahem, tried to take advantage of me when I was drunk." – He

      * *

Vice of Deficiency $\Rightarrow$ insensibility, or "too little" delight in
bodily pleasures

      * *

Virtue of Mean $\Rightarrow$ temperance, or "appropriate" delight (self-
control)

      * *

(X) Vice of Excess $\Rightarrow$ profligacy, or "too much" delight

  * •

(S3) GIVING (AND TAKING) - small amounts of money: how someone spends their
wealth on things whose worth are though of in terms of MONEY.

    * –

"Don’t ask Carla for any money, she would only lend it to you at large
interests." – Carla

      * *

(X) Vice of Deficiency $\Rightarrow$ stinginess, or "too little" giving (= too
much taking)

      * *

Virtue of Mean $\Rightarrow$ liberality/generosity, or "appropriate" giving
(spending on the appropriate things)

      * *

Vice of Excess $\Rightarrow$ prodigality/wastefulness, or "too much" giving (=
too little taking; squandering)

  * •

(S4) GIVING (AND TAKING) - other, added value: how someone spends their wealth
to create added value and to which things (their worth is not always
measurable in terms of money), spending/giving according to their wealth.

    * –

"Anna supports the animal shelter in our city financially and volunteers there
twice a week." – Anna

      * *

Vice of Deficiency $\Rightarrow$ meanness, or "too little" giving (little
added value)

      * *

(X) Virtue of Mean $\Rightarrow$ magnificence, or "appropriate" giving
(creating added value)

      * *

Vice of Excess $\Rightarrow$ vulgarity, or "too much" giving (making great
display on the wrong occasions and in the wrong way, little added value)

  * •

(S5) PRIDE (HONOUR = CAUSE): how someone communicates about and acts according
to one’s honour/pride. Honour is the CAUSE of their behavior.

    * –

"There are very low quality psychologists who go around trying to tell people
what to do and fix their problems but they are not qualified." – psychologists

      * *

Vice of Deficiency $\Rightarrow$ little-mindedness, or "too little" pride
(false modesty)

      * *

Virtue of Mean $\Rightarrow$ high-mindedness, or "appropriate" pride

      * *

(X) Vice of Excess $\Rightarrow$ vanity, or "too much" unjustified pride
(sense of superiority)

  * •

(S6) AMBITION (HONOUR = GOAL): How someone builds towards more honour and a
more virtuous self. Honour is the GOAL of their behavior.

    * –

"John’s parents had to force him to look for a job." – him (= John)

      * *

(X) Vice of Deficiency $\Rightarrow$ lack of ambition, or "too little"
ambition

      * *

Virtue of Mean $\Rightarrow$ proper ambition, or "appropriate" ambition

      * *

Vice of Excess $\Rightarrow$ over-ambitiousness, or "too much" ambition
(toxic)

  * •

(S7) ANGER: How someone deals with emotions, especially negative emotions such
as anger.

    * –

"Serena has a list of everyone who has ever wronged her." – Serena

      * *

Vice of Deficiency $\Rightarrow$ easy-tempered/spiritlessness, or "too little"
anger (emotionless, careless)

      * *

Virtue of Mean $\Rightarrow$ gentleness, "appropriate" anger (led by reason,
not by emotions)

      * *

(X) Vice of Excess $\Rightarrow$ hard-tempered, or "too much" anger
(wrathfulness, irritability)

  * •

(S8) PLEASURE (AND PAIN) OF OTHERS: general pleasantness in life, or how
someone agrees (pleasure of other) or disagrees (pain of other) with his/her
interlocutor in any type of interaction.

    * –

"My mother posted some of her poems on Facebook, but they were quickly filled
with mean and discouraging comments from many of her so-called friends." –
many of her so-called friends

      * *

(X) Vice of Deficiency $\Rightarrow$ cross/contentious, or "too little"
pleasure/agreement (= too much pain)

      * *

Virtue of Mean $\Rightarrow$ agreeableness/friendliness, or "appropriate" mix
of (dis)agreement

      * *

Vice of Excess $\Rightarrow$ flattery, or "too much" agreement (= too little
pain; people pleaser, often for own advantage)

  * •

(S9) TRUTH, HONESTY ABOUT ONESELF: How truthful someone is in their
interaction with others; what they say and how they say it.

    * –

"To me, it sounds weird to think that Jeremy is lying." – Jeremy

      * *

Vice of Deficiency $\Rightarrow$ irony, or "too little" truthfulness by
disclaiming/depreciating the truth (pretense as understatement)

      * *

(X) Virtue of Mean $\Rightarrow$ truthfulness, truthful in word and deed

      * *

Vice of Excess $\Rightarrow$ boastfulness, "too much" so-called truth
(pretense as exaggeration)

  * •

(S10) AMUSING CONVERSATION: How someone behaves during amusing conversation
and mixing with others.

    * –

"Hannah never says anything remotely funny, she’s always standing there with a
long face." – Hannah

      * *

(X) Vice of Deficiency $\Rightarrow$ boorishness, or "too little" amusement
(too boring, serious)

      * *

Virtue of Mean $\Rightarrow$ wittiness, "appropriate" amusement

      * *

Vice of Excess $\Rightarrow$ buffoonery, "too much" amusement (ridicule)

##  Appendix D Demographic Information on Human Evaluators

  * •

Gender: female, female

  * •

Age: 22, 23

  * •

Education level: master, master

  * •

Nationality: Belgian, Japanese


# An approximation of matrix exponential
by a truncated Laguerre series

E. D. Khoroshikh Department of System Analysis and Control, Voronezh State
University
1, Universitetskaya Square, Voronezh 394018, Russia<EMAIL_ADDRESS>and
V. G. Kurbatov∗ Department of System Analysis and Control, Voronezh State
University
1, Universitetskaya Square, Voronezh 394018, Russia<EMAIL_ADDRESS>
###### Abstract.

The Laguerre functions $l_{n,\tau}^{\alpha}$, $n=0,1,\dots$, are constructed
from generalized Laguerre polynomials. The functions $l_{n,\tau}^{\alpha}$
depend on two parameters: scale $\tau>0$ and order of generalization
$\alpha>-1$, and form an orthogonal basis in $L_{2}[0,\infty)$. Let the
spectrum of a square matrix $A$ lie in the open left half-plane. Then the
matrix exponential $H_{A}(t)=e^{At}$, $t>0$, belongs to $L_{2}[0,\infty)$.
Hence the matrix exponential $H_{A}$ can be expanded in a series
$H_{A}=\sum_{n=0}^{\infty}S_{n,\tau,\alpha,A}\,l_{n,\tau}^{\alpha}$. An
estimate of the norm
$\Bigl{\lVert}H_{A}-\sum_{n=0}^{N}S_{n,\tau,\alpha,A}\,l_{n,\tau}^{\alpha}\Bigr{\rVert}_{L_{2}[0,\infty)}$
is proposed. Finding the minimum of this estimate over $\tau$ and $\alpha$ is
discussed. Numerical examples show that the optimal $\alpha$ is often almost
0, which essentially simplifies the problem.

###### Key words and phrases:

matrix exponential, impulse response, generalized Laguerre polynomials,
Laguerre functions, estimate of accuracy, least square approximation, Wolfram
Mathematica

###### 1991 Mathematics Subject Classification:

Primary 65F60; Secondary 33C45, 97N50

∗ Corresponding author

## Introduction

An approximate calculation of the matrix exponential $H_{A}(t)=e^{At}$, $t>0$,
is of constant importance [7, 8, 10, 11, 15, 16] at least for solving linear
differential equations $\dot{x}(t)=Ax(t)+f(t)$. In this paper, the problem of
approximate representation of $H_{A}(t)$ in the from of a formula depending on
the parameter $t$ is discussed. The matrix $A$ is assumed to be stable, i. e.
the eigenvalues of $A$ lie in the open left half-plane.

Typically, the complete representation of $H_{A}(t)=e^{At}$ in the form of a
formula is very cumbersome (every element of the matrix function $H_{A}$ is a
linear combination of $M$ functions of the form $t\mapsto
t^{j}e^{\lambda_{k}t}$, where $M\times M$ is the size of the matrix $A$) and
therefore inconvenient for large $M$. We discuss the approximate
representation of $H_{A}$ in the form of a truncated Laguerre series; it
contains fewer terms and is therefore simpler than the complete $H_{A}$.

There are a lot of papers devoted to the approximation of impulse responses
and matrix exponential by the truncated Laguerre series, see, for example, [2,
4, 17, 19, 20, 21, 22, 24] and references therein. The main difference between
these papers and the present one lies in the formulation of the problem.

The _generalized Laguerre polynomials_ are the functions

$L_{n}^{\alpha}(t)=\frac{t^{-\alpha}e^{t}}{n!}\bigl{(}t^{n+\alpha}e^{-t}\bigr{)}^{(n)},\qquad\alpha>-1,\;t\geq
0,\;n=0,1,\dots.$

We call _Laguerre functions_ the modified Laguerre polynomials:

$l_{n,\tau}^{\alpha}(t)=\sqrt{\frac{n!}{\Gamma(n+\alpha+1)}}\tau^{\frac{\alpha+1}{2}}\,t^{\frac{\alpha}{2}}\,e^{-\tau
t/2}L_{n}^{\alpha}(\tau t),\qquad t\geq 0,\;\;n=0,1,\dots,$

The Laguerre functions depend on two parameters: the order of generalization
$\alpha>-1$ and the scale $\tau>0$. The most important and simple case is when
$\alpha=0$.

It is known that $l_{n,\tau}^{\alpha}$ form an orthonormal basis in
$L_{2}[0,\infty)$. Therefore the matrix exponential (impulse response)

$H_{A}(t)=e^{At},\qquad t>0,$

can be expanded in the _Laguerre series_

$H_{A}=\sum_{n=0}^{\infty}S_{n,\tau,\alpha,A}\,l_{n,\tau}^{\alpha}$

with matrix coefficients $S_{n,\tau,\alpha,A}$. The coefficient
$S_{n,\tau,\alpha,A}$ can be interpreted (Proposition 2) as the result of the
substitution of $A$ in the function $\lambda\mapsto
s_{n,\tau,\alpha,\lambda}$, where

$s_{n,\tau,\alpha,\lambda}=\int_{0}^{\infty}e^{\lambda
t}\,l_{n,\tau}^{\alpha}(t)\,dt.$

Since the matrix exponential $H_{A}$ is a linear combination of functions of
the form $t\mapsto t^{j}e^{\lambda_{k}t}$ (where $\lambda_{k}$ are eigenvalues
of $A$), which resemble $l_{n}$, it is natural to expect that the Laguerre
series converges rather fast and hence its truncation or partial sum

$H_{N,\tau,\alpha,A}=\sum_{n=0}^{N}S_{n,\tau,\alpha,A}\,l_{n,\tau}^{\alpha}$

approximates $H_{A}$ quite well.

The aim of this paper is estimating the accuracy $\lVert
H_{A}-H_{N,\tau,\alpha,A}\rVert_{L_{2}}$ and choosing $\tau$ and $\alpha$ that
provide the best estimate. The idea of the paper is as follows. We describe
(Theorem 4) the estimate of $\lVert H_{A}-H_{N,\tau,\alpha,A}\rVert_{L_{2}}$
in terms of the quantities

$\displaystyle\varphi(N,\tau,\alpha)=$
$\displaystyle\sum_{k=0}^{M}\zeta(N,\tau,\alpha,\lambda_{k}),$
$\displaystyle\psi(N,\tau,\alpha)=$
$\displaystyle\max_{k}\zeta(N,\tau,\alpha,\lambda_{k}),$

where $\lambda_{k}$, $k=1,2\dots,M$, are eigenvalues of $A$, and

$\zeta(N,\tau,\alpha,\lambda)=\int_{0}^{\infty}\Bigl{\lvert}e^{\lambda
t}-\sum_{n=0}^{N}\,s_{n,\tau,\alpha,\lambda}\,l_{n,\tau}^{\alpha}(t)\Bigr{\rvert}^{2}\,dt.$

We recommend to choose $\tau$ and $\alpha$ so that $\varphi(N,\tau,\alpha)$ be
minimal (alternatively, $\psi(N,\tau,\alpha)$ could be minimised). Numerical
experiments show that the optimal $\alpha$ is often close to 0. Optimization
with respect to $\tau$ only significantly simplifies and speeds up
calculations.

The paper is organized as follows. Sections 1 and 2 are devoted to definitions
and notation. In Section 3, we derive some formulas for Laguerre coefficients.
In Section 4, we describe the main estimate. In Section 5, we recall [2, 3, 4]
formulas for calculating the derivative of $\zeta$ with respect to $\tau$.
They simplifies the search for the minimum with respect to $\tau$ only. The
problem of finding minimum over $\tau$ only arises, for example, when we
restrict ourselves to the case $\alpha=0$; some simplified formulas for the
case $\alpha=0$ are collected in Section 6. In Section 7, we present the
recommending algorithm of finding $\tau$ and $\alpha$. In Section 8, we
describe the results of two numerical experiments.

We use ‘Wolfram Mathematica’ [27] for our computer calculations.

##  1\. The definition of Laguerre functions

In this section, we recall some definitions.

The functions

$\displaystyle L_{n}^{\alpha}(t)$
$\displaystyle=\frac{t^{-\alpha}e^{t}}{n!}\bigl{(}t^{n+\alpha}e^{-t}\bigr{)}^{(n)}$
$\displaystyle=\sum_{k=0}^{n}(-1)^{k}\binom{n+\alpha}{n-k}\frac{\lambda^{k}}{k!},\qquad
t\geq 0,\;\;\alpha>-1,\,n=0,1,\dots.$

are called [1, p. 775], [12, p. 71], [6, p. 31] _generalized Laguerre
polynomials_. The special cases

$L_{n}(t)=\frac{e^{t}}{n!}\bigl{(}t^{n}e^{-t}\bigr{)}^{(n)},\qquad t\geq
0,\;n=0,1,\dots.$

of these functions are called (_ordinary_) _Laguerre polynomials_. Actually,
$L_{n}^{\alpha}$ is a polynomial of degree $n$. It is well-known [23, Theorem
5.7.1], [14, p. 88], [18, § 8] that the functions $L_{n}^{\alpha}$ form an
orthogonal basis in $L_{2}[0,\infty)$ with the weight function $t\mapsto
t^{\alpha}e^{-t}$:

$\int_{0}^{\infty}t^{\alpha}e^{-t}L_{n}^{\alpha}(t)\,L_{m}^{\alpha}(t)\,dt=\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{nm},\qquad
n,m=0,1,\dots,$

where $\delta_{nm}$ is the Kronecker symbol. We note that the ordinary
polynomials $L_{n}$ are normalized, but the generalized ones $L_{n}^{\alpha}$,
$\alpha\neq 0$, are not.

Let $\tau>0$ be a given number. It plays a role of a time scale. We call the
family of functions

$\begin{split}l_{n,\tau}^{\alpha}(t)&=\sqrt{\frac{n!}{\Gamma(n+\alpha+1)}}\tau^{\frac{\alpha+1}{2}}\,t^{\frac{\alpha}{2}}\,e^{-\tau
t/2}L_{n}^{\alpha}(\tau t)\\\ &=\sqrt{\frac{\tau
n!}{\Gamma(n+\alpha+1)}}e^{-\tau
t/2}\sum_{k=0}^{n}(-1)^{k}\binom{n+\alpha}{n-k}\frac{(\tau\lambda)^{k+\alpha/2}}{k!},\quad
n=0,1,\dots.\end{split}$ (1)

the (_generalized_) _Laguerre functions_. In particular,

$l_{n,\tau}(t)=l_{n,\tau}^{0}(t)=\sqrt{\tau}\,e^{-\tau t/2}L_{n}(\tau
t),\qquad t\geq 0,\;\;n=0,1,\dots.$

Evidently, the Laguerre functions $l_{n,\tau}^{\alpha}$ form an orthonormal
basis in $L_{2}[0,\infty)$ (without weight):

$\int_{0}^{\infty}l_{n,\tau}^{\alpha}(t)\,l_{m,\tau}^{\alpha}(t)\,dt=\delta_{nm},\qquad
t\geq 0,\;\;n,m=0,1,\dots.$

##  2\. Laguerre series for $H_{A}$

In this section, we introduce some notation.

Let $M$ be a positive integer. We denote by $\mathbb{C}^{M\times M}$ the
linear space of all matrices of the size $M\times M$; the symbol
$\mathbf{1}\in\mathbb{C}^{M\times M}$ denotes the identical matrix.

Let $A\in\mathbb{C}^{M\times M}$ be a given matrix. We assume that $A$ is
_stable_ , i. e. the eigenvalues of $A$ lie in the open left half-plane. We
discuss the expansion of the function

$H_{A}(t)=e^{At},\qquad t>0,$

in the series of Laguerre functions. We call $H_{A}$ the _matrix exponential_
or the _impulse response_ of the differential equation

$\dot{x}(t)=Ax(t)+f(t).$

We recall that the generalized Laguerre functions (1) form an orthonormal
basis in $L_{2}[0,\infty)$; here the scale parameter $\tau>0$ and the order of
generalization $\alpha>-1$ can be chosen arbitrarily. Therefore the matrix
exponential $H_{A}$ can be represented in the form of the _Laguerre series_

$H_{A}=\sum_{n=0}^{\infty}S_{n,\tau,\alpha,A}\,l_{n,\tau}^{\alpha},$

where the Laguerre coefficients

$S_{n,\tau,\alpha,A}=\int_{0}^{\infty}H_{A}(t)\,l_{n,\tau}^{\alpha}(t)\,dt$
(2)

are matrices. Since the matrix exponential $H_{A}$ is a linear combination of
functions of the form $t\mapsto t^{j}e^{\lambda_{k}t}$, it is natural to
expect that the series converges rather fast and hence its $N$-truncation

$H_{N,\tau,\alpha,A}(t)=\sum_{n=0}^{N}S_{n,\tau,\alpha,A}\,l_{n,\tau}^{\alpha}(t)$
(3)

with relatively small $N$ approximates $H_{A}$ well enough. Nevertheless, note
that the number $N$ can not be taken very large, because of rounding errors.

The aim of this paper is to estimate the quantity

$\lVert
H_{A}-H_{N,\tau,\alpha,A}\rVert_{L_{2}[0,\infty)}=\sqrt{\int_{0}^{\infty}\bigl{\lVert}H_{A}(t)-H_{N,\tau,\alpha,A}(t)\bigr{\rVert}_{F}^{2}\,dt},$

where $\lVert\cdot\rVert_{F}$ is the Frobenius norm, and to give
recommendations on the optimal choice of $\tau$ and $\alpha$ based on it.

##  3\. The Laguerre coefficients of $h_{\lambda}$

In the simplest case, when the matrix $A$ has the size $1\times 1$, the
problem of construction of approximation (3) is reduced to the calculation of
Laguerre coefficients $s_{n,\tau,\alpha,\lambda}$ of the function $t\mapsto
e^{\lambda t}$; we do it in Proposition 1. Then we describe the expression of
$S_{n,\tau,\alpha,\lambda}$ in terms of $s_{n,\tau,\alpha,\lambda}$
(Proposition 2).

For $\operatorname{Re}\lambda<0$ (here and below $\operatorname{Re}$ means the
real part of a complex number), we consider the auxiliary function

$h_{\lambda}(t)=e^{\lambda t},\qquad t>0.$

It is straightforward to verify that

$\lVert
h_{\lambda}\rVert_{L_{2}[0,\infty)}=\frac{1}{\sqrt{-2\operatorname{Re}\lambda}}.$

Our interest in the function $h_{\lambda}$ is explained by the following. If
$\lambda$ is an eigenvalue of $A$ (recall that $\operatorname{Re}\lambda<0$)
and $v$ is the corresponding normalized eigenvector, then the function

$x_{\lambda}(t)=H_{A}(t)v$

can be represented as

$x_{\lambda}(t)=h_{\lambda}(t)v.$

Let us first perform some calculations with the functions $h_{\lambda}$. They
can be interpreted as the approximation of the matrix exponential $H_{A}$ by
the truncated Laguerre series (3) when $A$ is a matrix of the size $1\times 1$
whose only element equals $\lambda$.

We denote by $s_{n,\tau,\alpha,\lambda}$ the Laguerre coefficients of the
function $h_{\lambda}$ in the orthonormal basis $l_{n,\tau}^{\alpha}$:

$s_{n,\tau,\alpha,\lambda}=\int_{0}^{\infty}h_{\alpha,\lambda}(t)\,l_{n,\tau}^{\alpha}(t)\,dt,\qquad\operatorname{Re}\lambda<0.$
(4)

Clearly, $s_{n,\tau,\alpha,\lambda}$ are real for real $\lambda$. Therefore,
from the Schwarz reflection principle for holomorphic functions [9, theorem
7.5.2], it follows that

$\overline{s_{n,\tau,\alpha,\lambda}}=s_{n,\tau,\alpha,\bar{\lambda}},$ (5)

where the bar means the complex conjugate. Representation (5) is useful for
symbolic calculation of derivatives.

######  Proposition 1.

Let $\operatorname{Re}\lambda<0$. Then

$\begin{split}s_{n,\tau,\alpha,\lambda}&=\frac{\Gamma(\alpha/2+1)}{(\tau/2-\lambda)^{\alpha/2+1}}\,\tau^{\frac{\alpha+1}{2}}\,\binom{n+\alpha}{n}\,\sqrt{\frac{n!}{\Gamma(n+\alpha+1)}}\,\\\
&\times{}_{2}F_{1}\bigl{(}-n,\alpha/2+1,\alpha+1,\tau/(\tau/2-\lambda)\bigr{)},\end{split}$
(6)

where ${}_{2}F_{1}$ is the hypergeometric function. In particular,

$s_{n,\tau,0,\lambda}=-\frac{2\sqrt{\tau}(2\lambda+\tau)^{n}}{(2\lambda-\tau)^{n+1}},\qquad
n=0,1,\dots.$

######  Remark 1.

We note that the function
$z\mapsto{}_{2}F_{1}\bigl{(}-n,\alpha/2+1,\alpha+1,z\bigr{)}$ is a polynomial
of degree $n$, since [12, p. 10] its first argument $-n$ is a negative
integer. Thus it is calculated quickly and accurately.

###### Proof.

We begin with the formula [22, formula (16)]

$\displaystyle\int_{0}^{\infty}t^{\beta}\,e^{-\sigma t}\,L_{n}^{\alpha}(\tau
t)\,L_{k}^{\beta}(\sigma t)\,dt$
$\displaystyle=\binom{n+\alpha}{n-k}\binom{k+\beta}{k}\,\frac{\tau^{k}\,\Gamma(\beta+1)}{\sigma^{\beta+k+1}}$
$\displaystyle\times{}_{2}F_{1}\bigl{(}-n+k,\beta+k+1,\alpha+k+1,\tau/\sigma\bigr{)}.$

We have (see (1) and note that $L_{0}^{\beta}(t)=1$ for all $t\geq 0$ and
$\binom{\alpha/2}{0}=1$)

$\displaystyle s_{n,\tau,\alpha,\lambda}$
$\displaystyle=\int_{0}^{\infty}e^{\lambda
t}\,\sqrt{\frac{n!}{\Gamma(n+\alpha+1)}}\,\tau^{\frac{\alpha+1}{2}}\,t^{\frac{\alpha}{2}}\,e^{-\tau
t/2}\,L_{n}^{\alpha}(\tau t)\,dt$
$\displaystyle=\sqrt{\frac{n!}{\Gamma(n+\alpha+1)}}\,\tau^{\frac{\alpha+1}{2}}\,\int_{0}^{\infty}e^{(\lambda-\tau/2)t}\,t^{\frac{\alpha}{2}}\,L_{n}^{\alpha}(\tau
t)\,dt$
$\displaystyle=\sqrt{\frac{n!}{\Gamma(n+\alpha+1)}}\,\tau^{\frac{\alpha+1}{2}}\,\int_{0}^{\infty}t^{\frac{\alpha}{2}}\,e^{(\lambda-\tau/2)t}\,L_{n}^{\alpha}(\tau
t)\,L_{0}^{\alpha/2}\bigl{(}(\tau/2-\lambda)t\bigr{)}\,dt$
$\displaystyle=\sqrt{\frac{n!}{\Gamma(n+\alpha+1)}}\,\tau^{\frac{\alpha+1}{2}}\,\binom{n+\alpha}{n}\,\frac{\Gamma(\alpha/2+1)}{(\tau/2-\lambda)^{\alpha/2+1}}$
$\displaystyle\times{}_{2}F_{1}\bigl{(}-n,\alpha/2+1,\alpha+1,\tau/(\tau/2-\lambda)\bigr{)}.\qed$

######  Remark 2.

In a similar way one can derive the formula for the Laguerre coefficients of
the functions $t\mapsto t^{j}e^{\lambda t}$ which correspond to generalized
eigenvectors of $A$:

$\displaystyle q_{n,\tau,\alpha,\lambda}$
$\displaystyle=\int_{0}^{\infty}t^{j}e^{\lambda
t}\,l_{n,\tau}^{\alpha}(t)\,dt=\frac{\Gamma(\alpha/2+j+1)}{(\tau/2-\lambda)^{\alpha/2+1}}\,\tau^{\frac{\alpha+1}{2}}\,\binom{n+\alpha}{n}\,$
$\displaystyle\times\sqrt{\frac{n!}{\Gamma(n+\alpha+1)}}\,{}_{2}F_{1}\bigl{(}-n,\alpha/2+j+1,\alpha+1,\tau/(\tau/2-\lambda)\bigr{)}.$

######  Proposition 2.

Let the spectrum of $A$ lie in the open left half-plane. Then the coefficient
$S_{n,\tau,\alpha,A}$ is the function $\lambda\mapsto
s_{n,\tau,\alpha,\lambda}$ of $A$.

###### Proof.

Let $n\in\mathbb{N}_{0}$, $\tau>0$, and $\alpha>-1$ be fixed. For brevity, we
set $f(\lambda)=s_{n,\tau,\alpha,\lambda}$. From Proposition 1 it is seen that
$f$ is holomorphic in the open left half-plane $\operatorname{Re}\lambda<0$.
We recall that

$s_{n,\tau,\alpha,\lambda}=\int_{0}^{\infty}h_{\lambda}(t)\,l_{n,\tau}^{\alpha}(t)\,dt=\int_{0}^{\infty}e^{\lambda
t}\,l_{n,\tau}^{\alpha}(t)\,dt.$

From this formula, it is clear that

$\frac{\partial
s_{n,\tau,\alpha,\lambda}}{\partial\lambda}=t\,s_{n,\tau,\alpha,\lambda},\qquad\frac{\partial^{j}s_{n,\tau,\alpha,\lambda}}{\partial\lambda^{j}}=t^{j}\,s_{n,\tau,\alpha,\lambda}.$

We recall [5, ch. VII, § 1, Theorem 5], [13, ch. 1, § 5] that for any square
matrix $A$,

$f(A)=\sum_{k=1}^{m}\sum_{j=0}^{w_{k}-1}\frac{\partial^{j}f}{\partial\lambda^{j}}(\lambda_{k})\frac{N_{k}^{j}}{j!},$
(7)

where $\lambda_{k}$ are eigenvalues of $A$, $w_{k}$ are their multiplicities,
and $N_{k}$ are spectral nilpotents, in particular, $N_{k}^{0}=P_{k}$ are
spectral projectors. If all eigenvalues are simple, then

$f(A)=\sum_{k=1}^{M}f(\lambda_{k})P_{k}.$

For the exponential function $\lambda\mapsto e^{\lambda t}$ formula (7) takes
the form

$e^{At}=\sum_{k=1}^{m}\sum_{j=0}^{w_{k}-1}t^{j}e^{\lambda_{k}t}\frac{N_{k}^{j}}{j!}.$
(8)

In particular, if all eigenvalues are simple, then

$e^{At}=\sum_{k=1}^{M}e^{\lambda_{k}t}P_{k}.$

From (8) and (2) it follows that

$\displaystyle S_{n,\tau,\alpha,A}$
$\displaystyle=\int_{0}^{\infty}e^{At}\,l_{n,\tau}^{\alpha}(t)\,dt$
$\displaystyle=\int_{0}^{\infty}\sum_{k=1}^{m}\sum_{j=0}^{w_{k}-1}t^{j}e^{\lambda_{k}t}\frac{N_{k}^{j}}{j!}\,l_{n,\tau}^{\alpha}(t)\,dt$
$\displaystyle=\sum_{k=1}^{m}\sum_{j=0}^{w_{k}-1}\frac{N_{k}^{j}}{j!}\int_{0}^{\infty}t^{j}e^{\lambda_{k}t}\,l_{n,\tau}^{\alpha}(t)\,dt$
$\displaystyle=\sum_{k=1}^{m}\sum_{j=0}^{w_{k}-1}\frac{N_{k}^{j}}{j!}\,\frac{\partial^{j}s_{n,\tau,\alpha,\lambda_{k}}}{\partial\lambda^{j}},$

which, by (7), equals the function $\lambda\mapsto s_{n,\tau,\alpha,\lambda}$
of $A$. ∎

We do not discuss algorithms for calculating of the coefficients
$S_{n,\tau,\alpha,A}$ in this paper. One possibility for this purpose is
Proposition 2 and Corollary 9 (see below). When $S_{n,\tau,\alpha,A}$ are
found, we obtain the approximation

$H_{A}(t)\approx\sum_{n=0}^{N}S_{n,\tau,\alpha,A}\,l_{n,\tau}^{\alpha}(t).$

##  4\. The estimate of accuracy

In order for the truncated Laguerre series (2) approximate the matrix
exponential $H_{A}$ well enough, first of all, the truncated Laguerre series

$h_{N,\tau,\alpha,\lambda}=\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda}\,l_{n,\tau}^{\alpha}$

should approximate the function $h_{\lambda}$ for all $\lambda\in\sigma(A)$.
In this section, we discuss the inverse problem: how to estimate
$\bigl{\lVert}H_{A}-H_{N,\tau,\alpha,A}\bigr{\rVert}_{L_{2}}$ in terms of
$\bigl{\lVert}h_{\lambda_{k}}-h_{N,\tau,\alpha,\lambda_{k}}\bigr{\rVert}_{L_{2}}$,
where $\lambda_{k}$ runs over the eigenvalues of $A$.

For $\operatorname{Re}\lambda<0$ and a natural number $N$ we denote by
$\zeta(N,\tau,\alpha,\lambda)$ the square of the accuracy of the approximation
of the function $h_{\lambda}$ by its $N$-truncated Laguerre series:

$\zeta(N,\tau,\alpha,\lambda)=\int_{0}^{\infty}\Bigl{\lvert}e^{\lambda
t}-\sum_{n=0}^{N}\,s_{n,\tau,\alpha,\lambda}\,l_{n,\tau}^{\alpha}(t)\Bigr{\rvert}^{2}\,dt.$
(9)

Clearly, we can rewrite this formula as

$\begin{split}\zeta(N,\tau,\alpha,\lambda)&=\Bigl{\lVert}h_{\lambda}-\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda}\,l_{n,\tau}^{\alpha}\Bigr{\rVert}_{L_{2}}^{2}\\\
&=\Bigl{\lVert}\sum_{n=N+1}^{\infty}s_{n,\tau,\lambda}\,l_{n,\tau}\Bigr{\rVert}^{2}\\\
&=\sum_{n=N+1}^{\infty}|s_{n,\tau,\alpha,\lambda}|^{2}.\end{split}$ (10)

For a matrix $C=\\{C_{ij}\\}\in\mathbb{C}^{M\times M}$, we denote by $\lVert
C\rVert_{2\to 2}$ the norm induced by the Euclidean norm
$\lVert\cdot\rVert_{2}$ on $\mathbb{C}^{M}$ and by

$\lVert C\rVert_{F}=\sqrt{\sum_{i=1}^{M}\sum_{j=1}^{M}|C_{ij}|^{2}}$

the Frobenius norm [8, p. 71]. It is easy to show that

$\displaystyle\lVert A\rVert_{2\to 2}$ $\displaystyle\leq\lVert A\rVert_{F},$
$\displaystyle\lVert AB\rVert_{F}$ $\displaystyle\leq\lVert A\rVert_{2\to
2}\cdot\lVert B\rVert_{F},$ $\displaystyle\lVert AB\rVert_{F}$
$\displaystyle\leq\lVert A\rVert_{F}\cdot\lVert B\rVert_{2\to 2},$
$\displaystyle\lVert Ax\rVert_{2}$ $\displaystyle\leq\lVert
A\rVert_{F}\cdot\lVert x\rVert_{2}.$

By default, we use for matrices $C\in\mathbb{C}^{M\times M}$ the Frobenius
norm. We denote by $\sigma(C)$ the spectrum (the set of all eigenvalues) of a
square matrix $C$. We recall that eigenvalues and eigenvectors are calculated
[27] with high backward stability by the QR-algorithm [8, 10, 26], so we
suppose that they are known.

######  Proposition 3.

Let $D\in\mathbb{C}^{M\times M}$ be a diagonal matrix with diagonal elements
$\lambda_{k}$, $\operatorname{Re}\lambda_{k}<0$, $k=1,2,\dots,M$. Then for the
number

$\lVert
H_{D}-H_{N,\tau,\alpha,D}\rVert_{L_{2}}=\sqrt{\int_{0}^{\infty}\bigl{\lVert}H_{D}(t)-H_{N,\tau,\alpha,D}(t)\bigr{\rVert}_{F}^{2}\,dt},$

we have

$\lVert
H_{D}-H_{N,\tau,\alpha,D}\rVert_{L_{2}}=\sqrt{\sum_{k=0}^{M}\zeta(N,\tau,\alpha,\lambda_{k})}\leq\sqrt{M\max_{k}\zeta(N,\tau,\alpha,\lambda_{k})},$

where $\lambda_{k}$ are the eigenvalues of $D$ and the function $\zeta$ is
defined by (9).

###### Proof.

By assumption, the matrix $D$ has the form

$D=\begin{pmatrix}\lambda_{1}&0&\ldots&0\\\ 0&\lambda_{2}&\ldots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\ldots&\lambda_{M}\end{pmatrix}.$

Therefore,

$H_{D}(t)=\begin{pmatrix}h_{\lambda_{1}}(t)&0&\ldots&0\\\
0&h_{\lambda_{2}}(t)&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&h_{\lambda_{M}}(t)\end{pmatrix}$

and

$H_{N,\tau,\alpha,D}(t)\\\
=\begin{pmatrix}\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda_{1}}\,l_{n,\tau}^{\alpha}(t)&0&\ldots&0\\\
0&\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda_{2}}\,l_{n,\tau}^{\alpha}(t)&\ldots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda_{M}}\,l_{n,\tau}^{\alpha}(t)\end{pmatrix}.$

Hence, by the definition of the Frobenius norm,

$\bigl{\lVert}H_{D}(t)-H_{N,\tau,\alpha,D}(t)\bigr{\rVert}_{F}^{2}=\sum_{k=0}^{M}\Bigl{\lvert}h_{\lambda_{k}}(t)-\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda_{k}}\,l_{n,\tau}^{\alpha}(t)\Bigr{\rvert}^{2}.$

Consequently, (recall that $\operatorname{Re}\lambda_{k}<0$)

$\sqrt{\int_{0}^{\infty}\bigl{\lVert}H_{D}(t)-H_{N,\tau,\alpha,D}(t)\bigr{\rVert}_{F}^{2}\,dt}\\\
=\sqrt{\int_{0}^{\infty}\sum_{k=0}^{M}\Bigl{\lvert}h_{\lambda_{k}}(t)-\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda_{k}}\,l_{n,\tau}^{\alpha}(t)\Bigr{\rvert}^{2}\,dt}\\\
=\sqrt{\sum_{k=0}^{M}\int_{0}^{\infty}\Bigl{\lvert}h_{\lambda_{k}}(t)-\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda_{k}}\,l_{n,\tau}^{\alpha}(t)\Bigr{\rvert}^{2}\,dt}\\\
=\sqrt{\sum_{k=0}^{M}\zeta(N,\tau,\alpha,\lambda_{k})}.\qed$

Now let us suppose that the matrix $A$ is _diagonalizable_ ; this means that
there exists an invertible matrix $T$ and a diagonal matrix $D$ such that

$A=TDT^{-1}.$

In such a case, the diagonal elements of $D$ are the eigenvalues of $A$ and
the columns of $T$ are the corresponding eigenvectors. Without loss of
generality we can assume that the columns of $T$ have unit Euclidian norm. The
matrix $D$ can be interpreted as the Jordan form of the matrix $A$; thus, a
diagonalizable matrix has (complex) Jordan blocks of the size $1\times 1$
only. It is clear that for a diagonalizable matrix $A$,

$\displaystyle H_{A}(t)$ $\displaystyle=e^{At}=Te^{Dt}T^{-1},$ $\displaystyle
t$ $\displaystyle>0,$ $\displaystyle H_{N,\tau,\alpha,A}(t)$
$\displaystyle=\sum_{n=0}^{N}TS_{n,\tau,\alpha,D}T^{-1}\,l_{n,\tau}^{\alpha}(t),$
$\displaystyle t$ $\displaystyle>0.$

Recall that we use the Frobenius norm in the space $\mathbb{C}^{M\times M}$.

######  Theorem 4.

Let $A\in\mathbb{C}^{M\times M}$ and $\lambda_{k}$, $k=1,2,\dots,M$, be
eigenvalues of $A$. Then

$\sqrt{\max_{k}\zeta(N,\tau,\alpha,\lambda_{k})}\leq\lVert
H_{A}-H_{N,\tau,\alpha,A}\rVert_{L_{2}[0,\infty)}$ (11)

and (provided that $A$ is diaginalizable)

$\begin{split}\lVert
H_{A}-H_{N,\tau,\alpha,A}\rVert_{L_{2}[0,\infty)}&\leq\varkappa(T)\sqrt{\sum_{k=0}^{M}\zeta(N,\tau,\alpha,\lambda_{k})}\\\
&\leq\varkappa(T)\sqrt{M\max_{k}\zeta(N,\tau,\alpha,\lambda_{k})},\end{split}$
(12)

where $\varkappa(T)=\lVert T\rVert_{2\to 2}\cdot\lVert T^{-1}\rVert_{2\to 2}$
is the condition number [10, p. 63] of $T$.

###### Proof.

Let $\lambda$ be an eigenvalue of $A$ and $v$ be the corresponding normalized
eigenvector. Since $H_{A}(t)$ and $S_{n,\tau,\alpha,\lambda}$ are respectively
the functions $\lambda\mapsto h_{\lambda}(t)$ and $\lambda\mapsto
s_{n,\tau,\alpha,\lambda}$ of $A$ (Proposition 2), $v$ is also the eigenvector
of $H_{A}(t)$ and $S_{n,\tau,\alpha,\lambda}$, and it corresponds to the
eigenvalues $h_{\lambda}(t)$ and $s_{n,\tau,\alpha,\lambda}$:

$\displaystyle H_{A}(t)v$ $\displaystyle=h_{\lambda}(t)v,$ $\displaystyle
H_{N,\tau,\alpha,A}(t)v$
$\displaystyle=\Bigl{(}\sum_{n=0}^{N}S_{n,\tau,\alpha,\lambda}\,l_{n,\tau}^{\alpha}(t)\Bigr{)}v=\Bigl{(}\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda}\,l_{n,\tau}^{\alpha}(t)\Bigr{)}v.$

Therefore,

$\displaystyle\bigl{\lVert}H_{A}-H_{N,\tau,\alpha,A}\bigr{\rVert}_{L_{2}}$
$\displaystyle\geq\bigl{\lVert}(H_{A}-H_{N,\tau,\alpha,A})v\bigr{\rVert}_{L_{2}}$
$\displaystyle=\sqrt{\int_{0}^{\infty}\bigl{\lVert}\bigl{(}H_{A}(t)-H_{N,\tau,\alpha,A}(t)\bigr{)}v\bigr{\rVert}^{2}\,dt}$
$\displaystyle=\sqrt{\int_{0}^{\infty}\Bigl{\lVert}\Bigl{(}h_{\lambda}(t)-\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda}\,l_{n,\tau}^{\alpha}(t)\Bigr{)}v\Bigr{\rVert}^{2}\,dt}$
$\displaystyle=\sqrt{\int_{0}^{\infty}\Bigl{\lvert}h_{\lambda}(t)-\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda}(t)\,l_{n,\tau}^{\alpha}\Bigr{\rvert}^{2}\cdot\lVert
v\rVert^{2}\,dt}$
$\displaystyle=\sqrt{\int_{0}^{\infty}\Bigl{\lvert}h_{\lambda}(t)-\sum_{n=0}^{N}s_{n,\tau,\alpha,\lambda}(t)\,l_{n,\tau}^{\alpha}\Bigr{\rvert}^{2}}\,dt$
$\displaystyle=\sqrt{\zeta(N,\tau,\alpha,\lambda)}.$

From this inequality, it follows estimate (11).

Estimate (12) follows from Proposition 3 and the inequality

$\lVert TBT^{-1}\rVert_{F}\leq\lVert T\rVert_{2\to 2}\cdot\lVert
B\rVert_{F}\cdot\lVert T^{-1}\rVert_{2\to 2}=\varkappa(T)\cdot\lVert
B\rVert_{F}.\qed$

##  5\. Derivatives with respect to $\tau$

Derivatives with respect to $\tau$ of some of the involved functions have
simple representations. This can help to find extreme points. In this section,
we present relevant statements.

######  Proposition 5 (see [2, 3, 4]).

We have

$\frac{\partial
l_{n,\tau}^{\alpha}}{\partial\tau}(t)=d_{n+1}\,l_{n+1,\tau}^{\alpha}(t)-d_{n}l_{n-1,\tau}^{\alpha}(t),\qquad
n=0,1,\dots,$

where $l_{-1,\tau}^{\alpha}(t)=0$ and

$d_{0}=0,\qquad d_{n}=\frac{\sqrt{n(n+\alpha)}}{2\tau}.$

In particular, for $\alpha=0$,

$d_{n}=\frac{n}{2\tau}.$

###### Proof.

The proof follows from (1) and the well-known [23, formula (5.1.14)] formula

$\frac{\partial L_{n,t}^{\alpha}}{\partial t}=-L_{n-1,t}^{\alpha+1}.\qed$

######  Corollary 6.

For Laguerre coefficients (6), we have

$\frac{\partial
s_{n,\tau,\alpha,\lambda}}{\partial\tau}=d_{n+1}\,s_{n+1,\tau,\alpha,\lambda}-d_{n}\,s_{n-1,\tau,\alpha,\lambda},\qquad
n=0,1,\dots.$

###### Proof.

It follows directly from (4) and Proposition 5. ∎

######  Corollary 7.

For function (9), we have

$\displaystyle\frac{\partial\zeta(N,\tau,\alpha,\lambda)}{\partial\tau}$
$\displaystyle=-d_{N+1}\bigl{(}s_{N+1,\tau,\alpha,\lambda}\,s_{N,\tau,\alpha,\bar{\lambda}}+s_{N+1,\tau,\alpha,\bar{\lambda}}\,s_{N,\tau,\alpha,\lambda}\bigr{)}$
$\displaystyle=-2d_{N+1}\operatorname{Re}\bigl{(}s_{N+1,\tau,\alpha,\lambda}\,s_{N,\tau,\alpha,\bar{\lambda}}\bigr{)},$

where the bar over $\lambda$ means complex conjugate of $\lambda$.

###### Proof.

We make use of representations (10) and (5):

$\zeta(N,\tau,\alpha,\lambda)=\sum_{n=N+1}^{\infty}|s_{n,\tau,\alpha,\lambda}|^{2}=\sum_{n=N+1}^{\infty}s_{n,\tau,\alpha,\lambda}\,\overline{s_{n,\tau,\alpha,\lambda}}=\sum_{n=N+1}^{\infty}s_{n,\tau,\alpha,\lambda}\,s_{n,\tau,\alpha,\bar{\lambda}}.$

Differentiating the last formula, we obtain

$\frac{\partial\zeta(N,\tau,\alpha,\lambda}{\partial\tau}=\sum_{n=N+1}^{\infty}\frac{\partial
s_{n,\tau,\alpha,\lambda}}{\partial\tau}\,s_{n,\tau,\alpha,\bar{\lambda}}+s_{n,\tau,\alpha,\lambda}\,\frac{\partial
s_{n,\tau,\alpha,\bar{\lambda}}}{\partial\tau}.$

Then from Corollary 6 it follows

$\displaystyle\frac{\partial\zeta(N,\tau,\alpha,\lambda}{\partial\tau}$
$\displaystyle=\sum_{n=N+1}^{\infty}\bigl{(}d_{n+1}\,s_{n+1,\tau,\alpha,\lambda}-d_{n}\,s_{n-1,\tau,\alpha,\lambda}\bigr{)}\,s_{n,\tau,\alpha,\bar{\lambda}}$
$\displaystyle+s_{n,\tau,\alpha,\lambda}\,\bigl{(}d_{n+1}\,s_{n+1,\tau,\alpha,\bar{\lambda}}-d_{n}\,s_{n-1,\tau,\alpha,\bar{\lambda}}\bigr{)}$
$\displaystyle=\sum_{n=N+1}^{\infty}\bigl{(}d_{n+1}\,s_{n+1,\tau,\alpha,\lambda}\,s_{n,\tau,\alpha,\bar{\lambda}}-d_{n}\,s_{n,\tau,\alpha,\bar{\lambda}}\,s_{n-1,\tau,\alpha,\lambda}$
$\displaystyle+d_{n+1}\,s_{n+1,\tau,\alpha,\bar{\lambda}}\,s_{n,\tau,\alpha,\lambda}-d_{n}\,s_{n,\tau,\alpha,\lambda}\,s_{n-1,\tau,\alpha,\bar{\lambda}}\bigr{)}.$

After canceling we obtain the desired representation. ∎

##  6\. The case $\alpha=0$

Our numerical experiments (see Section 8) show that often the optimal value of
$\alpha$ is close to 0. For this reason, we treated the case of $\alpha=0$ as
a special one in the previous exposition. In this section, we collect some
additional formulas related to $\alpha=0$. These formulas and Corollary 7
allow one to organize calculations for the case $\alpha=0$ substantially
simpler and faster than for the general case. Thus, by taking $\alpha$ to be
equal 0 (though the optimal $\alpha$ is only close to 0), we can take a larger
number $N$ of terms in the truncated Laguerre series (3) and thereby
compensate for the small loss of accuracy caused by a notoptimal value of
$\alpha$.

######  Proposition 8.

Let $\operatorname{Re}\lambda<0$. Then the Laguerre coefficients
$s_{n,\tau,0,\lambda}$ can be calculated recursively:

$\displaystyle s_{0,\tau,0,\lambda}$
$\displaystyle=-\frac{2\sqrt{\tau}}{2\lambda-\tau},$ $\displaystyle
s_{n+1,\tau,0,\lambda}$
$\displaystyle=\frac{2\lambda+\tau}{2\lambda-\tau}\,s_{n,\tau,0,\lambda}.$

###### Proof.

It follows from Proposition 1. ∎

######  Corollary 9.

Let the spectrum of $A$ lie in the left half-plane. Then the Laguerre
coefficients $S_{n,\tau,0,A}$ can be calculated recursively:

$\displaystyle S_{0,\tau,0,A}$
$\displaystyle=-2\sqrt{\tau}(2A-\tau\mathbf{1})^{-1},$ $\displaystyle
S_{n+1,\tau,0,A}$
$\displaystyle=(2A+\tau\mathbf{1})(2A-\tau\mathbf{1})^{-1}\,S_{n,\tau,0,A}.$

###### Proof.

The proof follows from Propositions 8 and 2. ∎

It is convenient to use Corollary 9 for calculating $S_{n,\tau,0,A}$.

######  Corollary 10.

Let $\operatorname{Re}\lambda<0$. Then

$\displaystyle\zeta(N,\tau,0,\lambda)$
$\displaystyle=\frac{4\tau}{|2\lambda-\tau|^{2}}\cdot\frac{\Bigl{|}\dfrac{2\lambda+\tau}{2\lambda-\tau}\Bigr{|}^{2N+2}}{1-\Bigl{|}\dfrac{2\lambda+\tau}{2\lambda-\tau}\Bigr{|}^{2}}$
$\displaystyle=\frac{4\tau}{|2\lambda-\tau|^{2}-|2\lambda+\tau|^{2}}\Bigl{|}\frac{2\lambda+\tau}{2\lambda-\tau}\Bigr{|}^{2N+2}.$

###### Proof.

The proof immediately follows from Proposition 8. ∎

##  7\. The choice of $\tau$ and $\alpha$

In this section, we propose an algorithm for choosing $\tau$ and $\alpha$.

Let a stable matrix $A$ be given. By means of the Jordan decomposition, we
calculate eigenvalues and eigenvectors of $A$. Typically, at least due to
rounding errors, the spectrum of $A$ is simple, moreover, all eigenvalues
$\lambda_{k}$ are distinct. If the spectrum of $A$ is not simple, the proposed
algorithm for choosing $\tau$ and $\alpha$ also works, but less can be said
about the approximation accuracy.

We take a number $N\in\mathbb{N}$. For example, we take $N=10$. We do not use
$N>50$ because of large rounding errors.

We consider the function

$\varphi(N,\tau,\alpha)=\sum_{k=0}^{M}\zeta(N,\tau,\alpha,\lambda_{k}),$

where $\lambda_{k}$ are the eigenvalues of $A$ and $\zeta$ is defined by (9).

First, we consider the case $\alpha=0$. By Corollary 7, we have

$\frac{\partial\zeta(N,\tau,0,\lambda}{\partial\tau}=-2d_{N+1}\operatorname{Re}\bigl{(}s_{N+1,\tau,0,\lambda}\,s_{N,\tau,0,\bar{\lambda}}\bigr{)}.$

From Proposition 1 we know that

$s_{n,\tau,0,\lambda}=-\frac{2\sqrt{\tau}(2\lambda+\tau)^{n}}{(2\lambda-\tau)^{n+1}},\qquad
n=0,1,\dots.$

Therefore,

$\frac{\partial\zeta(N,\tau,0,\lambda}{\partial\tau}=-2d_{N+1}\operatorname{Re}\Bigl{(}\frac{2\sqrt{\tau}(2\lambda+\tau)^{N+1}}{(2\lambda-\tau)^{N+2}}\,\frac{2\sqrt{\tau}(2\bar{\lambda}+\tau)^{N}}{(2\bar{\lambda}-\tau)^{N+1}}\Bigr{)}.$

Finally, we arrive at

$\frac{\partial\varphi(N,\tau,0)}{\partial\tau}=-2d_{N+1}\sum_{k=0}^{M}\operatorname{Re}\Bigl{(}\frac{2\sqrt{\tau}(2\lambda_{k}+\tau)^{N+1}}{(2\lambda_{k}-\tau)^{N+2}}\,\frac{2\sqrt{\tau}(2\bar{\lambda}_{k}+\tau)^{N}}{(2\bar{\lambda}_{k}-\tau)^{N+1}}\Bigr{)}.$

Numerical experiments show that the function
$\tau\mapsto\frac{\partial\varphi(N,\tau,0)}{\partial\tau}$ increases. Hence
the function $\tau\mapsto\varphi(N,\tau,0)$ has a unique minimum. We find it
by solving the equation

$\frac{\partial\varphi(N,\tau,0)}{\partial\tau}=0$

for $\tau$ (in ‘Mathematica’ [27] it is done by the command `FindRoot`; this
command works iteratively; we take for the initial value $\tau=1$). It can be
done quite quickly for values of $N$ up to 50. Thus, we find the optimal
$\tau$ for the case $\alpha=0$. Let us denote it by $\tau_{0}$. After that we
calculate $\varphi(N,\tau_{0},0)$ using Corollary 10 and the definition of
$\varphi$.

Then we calculate symbolically $\varphi(N,\tau,\alpha)$ using formulas (6) and
(10). Of course, the resulting formula is rather cumbersome. We calculate (in
‘Mathematica’ [27] it is done by the command `FindMinimum`)

$\varphi_{\min}(N)=\min_{\tau>0,\,\alpha>-1}\varphi(N,\tau,\alpha).$

We take only $N\leq 12$, because the calculations are notedly slow for greater
$N$. We take the found point of minimum $(\tau_{1},\alpha_{1})$ as the optimal
values of $\tau$ and $\alpha$. We use $\varphi_{\min}$ for the estimates of
$\lVert H_{A}-H_{N,\tau,\alpha,A}\rVert_{L_{2}[0,\infty)}$ according to
Theorem 4; for the same aim, we also calculate

$\psi(N,\tau_{1},\alpha_{1})=\max_{\lambda_{k}\in\sigma(A)}\zeta(N,\tau_{1},\alpha_{1},\lambda_{k})$

for the found $\tau_{1}$ and $\alpha_{1}$.

Our numerical experiments (see Section 8) show that the pair $\tau_{0}$ and
$\alpha_{0}=0$ is often almost optimal. So, the consideration of $\alpha\neq
0$ is not always necessarily.

##  8\. Numerical experiments

In this section, we describe two numerical examples.

######  Example 1.

We consider the discrete model of a transmission line shown in fig 2. We
assume that the line consists of $n=150$ sections. Thus we have 150 unknown
currents $I_{C}$ and 150 unknown voltages $U_{L}$. The parameters are as
follows: $C=C_{0}/n$, $L=L_{0}/n$, $R=R_{0}/n$, $G=G_{0}/n$, where $C_{0}=10$,
$L_{0}=50$, $R_{0}=170$, $G_{0}=160$. The modified nodal [25] description of
the circuit has the form $\dot{x}(t)=Ax(t)+f(t)$ with a matrix $A$ of the size
$300\times 300$. The spectrum of $-A$ is shown in fig. 2.

First, we take $N=10$ in the truncated Laguerre series (3). We calculate the
minimum of $\varphi$ over $\tau$ for $\alpha=0$; as the initial value of
$\tau$ we take $\tau=1$. We obtain the following results (see the left fig.
2). The optimal $\tau$ is $\tau_{0}=19.2$ (it is shown on the left fig. 2 as a
small square); $\sqrt{\varphi(10,\tau_{0},0)}=0.00104$; $\varkappa(T)=28.4$;
$\sqrt{M}=17.3$. According to Theorem 4 we have

$\displaystyle\lVert H_{A}-H_{10,\tau_{0},0,A}\rVert_{L_{2}[0,\infty)}$
$\displaystyle\geq\sqrt{\psi(10,\tau_{0},0)}=0.000192,$ $\displaystyle\lVert
H_{A}-H_{10,\tau_{0},0,A}\rVert_{L_{2}[0,\infty)}$
$\displaystyle\leq\varkappa(T)\,\sqrt{\varphi(10,\tau_{0},0)}=0.0294,$
$\displaystyle\lVert H_{A}-H_{10,\tau_{0},0,A}\rVert_{L_{2}[0,\infty)}\ $
$\displaystyle\leq\varkappa(T)\sqrt{M\,\psi(10,\tau_{0},0)}=0.0945.$

Second, we repeat the same experiment with $N=30$. The result is as follows:
$\tau_{0}$ is $19.3$, but $\sqrt{\varphi(30,\tau_{0},0)}=4.21\cdot 10^{-8}$,
and

$\displaystyle\lVert H_{A}-H_{30,\tau_{0},0,A}\rVert_{L_{2}[0,\infty)}$
$\displaystyle\geq\sqrt{\psi(30,\tau_{0},0)}=8.33\cdot 10^{-9},$
$\displaystyle\lVert H_{A}-H_{30,\tau_{0},0,A}\rVert_{L_{2}[0,\infty)}$
$\displaystyle\leq\varkappa(T)\,\sqrt{\varphi(30,\tau_{0},0)}=1.83\cdot
10^{-7},$ $\displaystyle\lVert
H_{A}-H_{30,\tau_{0},0,A}\rVert_{L_{2}[0,\infty)}\ $
$\displaystyle\leq\varkappa(T)\sqrt{M\,\psi(30,\tau_{0},0)}=4.09\cdot
10^{-6}.$

Third, we return to $N=10$, take as initial values the found $\tau_{0}=19.2$
and $\alpha=0$, and find the minimum of $\varphi(N,\tau,\alpha)$ over $\tau$
and $\alpha$. We obtain the following results (right fig. 2). The optimal
$\tau$ is $\tau_{1}=19.20$; the optimal $\alpha$ is $\alpha_{1}=0.0000239$;
$\sqrt{\varphi(10,\tau_{1},\alpha_{1})}=0.001036$; $\varkappa(T)=28.358$;
$\sqrt{M}=17.3$. According to Theorem 4 we have

$\displaystyle\lVert
H_{A}-H_{10,\tau_{1},\alpha_{1},A}\rVert_{L_{2}[0,\infty)}$
$\displaystyle\geq\sqrt{\psi(10,\tau_{1},\alpha_{1})}=0.000193,$
$\displaystyle\lVert
H_{A}-H_{10,\tau_{1},\alpha_{1},A}\rVert_{L_{2}[0,\infty)}$
$\displaystyle\leq\varkappa(T)\,\sqrt{\varphi(10,\tau_{1},\alpha_{1})}=0.0294,$
$\displaystyle\lVert
H_{A}-H_{10,\tau_{1},\alpha_{1},A}\rVert_{L_{2}[0,\infty)}\ $
$\displaystyle\leq\varkappa(T)\sqrt{M\,\psi(10,\tau_{1},\alpha_{1})}=0.0947.$

Figure 1. A discrete model of a transmission line

Figure 2. The points show the spectrum of the matrix $-A$ from example 1, the
small squares are the found $\tau$. In the right figure the minimum is taken
over $\tau$ and $\alpha$; in the left figure the minimum is taken only over
$\tau$ with $\alpha=0$

######  Example 2.

In Example 1 the minimum of $\varphi(N,\tau,\alpha)$ is attained almost at
$\alpha=0$. We present here another example of the same kind. Since our
estimate depends only on the spectrum of $A$, we do not present a matrix $A$
itself and work only with its possible spectrum.

We take 200 random complex numbers; their real parts have the Maxwell
distribution with $\sigma=4$ (the probability density for value $x$ in the
Maxwell distribution is proportional to $x^{2}e^{-x^{2}/(2\sigma^{2})}$ for
$x>0$, and is zero for $x<0$) and imaginary parts have the normal distribution
with the mean value $\mu=0$ and the variance $\sigma^{2}=1$. We interpret
these points as a possible spectrum of $-A$; we present them in fig. 4. The
results of calculation are as follows.

First we take $N=10$ and $\alpha=0$. Starting from the initial point $\tau=1$
we find that the minimum of $\varphi(10,\tau,0)$ over $\tau$ is attained at
$\tau_{0}=5.83$ and

$\displaystyle\sqrt{\varphi(10,\tau_{0},0)}$ $\displaystyle=0.00878,$
$\displaystyle\sqrt{\psi(10,\tau_{0},0)}$ $\displaystyle=0.00742.$

The experiment with $N=30$ and $\alpha=0$ gives $\tau_{0}=5.67$ and

$\displaystyle\sqrt{\varphi(30,\tau_{0},0)}$ $\displaystyle=1.58\cdot
10^{-6},$ $\displaystyle\sqrt{\psi(30,\tau_{0},0)}$ $\displaystyle=1.32\cdot
10^{-6}.$

Then we take $N=10$ and begin iterations from the found $\tau_{0}=5.83$ and
$\alpha=0$. Now the optimal $\tau$ is $\tau_{1}=5.82$ and the optimal $\alpha$
is $\alpha_{1}=-0.000915$. For the estimates from Theorem 4 we have

$\displaystyle\sqrt{\varphi(10,\tau_{1},\alpha_{1})}$ $\displaystyle=0.00874,$
$\displaystyle\sqrt{\psi(10,\tau_{1},\alpha_{1})}$ $\displaystyle=0.00738.$

Figure 3. The points from example 2, the small squares are the found $\tau$.
In the right figure the minimum is taken over $\tau$ and $\alpha$; in the left
figure the minimum is taken only over $\tau$ with $\alpha=0$. Note that in the
right fig. $N=10$, but in the left fig. $N=30$

.

Figure 4. The graphs of the function $\varphi$ from Examples 1 (left) and 2
(right) for $N=10$ in a neighbourhood of the minimum point

Fig. 4 shows that the function $\varphi$ is rather smooth and convex. Thus,
the problem of finding of its minimum can be solved by standard tools.

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